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Properties

Label 2-115-115.102-c1-0-9
Degree 22
Conductor 115115
Sign −0.951+0.307i-0.951 + 0.307i
Analytic cond. 0.9182790.918279
Root an. cond. 0.9582690.958269
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.174 − 2.43i)2-s + (−0.591 + 0.128i)3-s + (−3.92 − 0.564i)4-s + (0.343 − 2.20i)5-s + (0.210 + 1.46i)6-s + (0.0839 − 0.0458i)7-s + (−1.02 + 4.69i)8-s + (−2.39 + 1.09i)9-s + (−5.32 − 1.22i)10-s + (2.37 + 2.06i)11-s + (2.39 − 0.171i)12-s + (3.02 − 5.54i)13-s + (−0.0970 − 0.212i)14-s + (0.0810 + 1.35i)15-s + (3.64 + 1.07i)16-s + (3.26 + 2.44i)17-s + ⋯
L(s)  = 1  + (0.123 − 1.72i)2-s + (−0.341 + 0.0742i)3-s + (−1.96 − 0.282i)4-s + (0.153 − 0.988i)5-s + (0.0858 + 0.597i)6-s + (0.0317 − 0.0173i)7-s + (−0.361 + 1.65i)8-s + (−0.798 + 0.364i)9-s + (−1.68 − 0.386i)10-s + (0.716 + 0.621i)11-s + (0.691 − 0.0494i)12-s + (0.839 − 1.53i)13-s + (−0.0259 − 0.0567i)14-s + (0.0209 + 0.348i)15-s + (0.912 + 0.267i)16-s + (0.791 + 0.592i)17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((−0.951+0.307i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+1/2) L(s)=((−0.951+0.307i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: −0.951+0.307i-0.951 + 0.307i
Analytic conductor: 0.9182790.918279
Root analytic conductor: 0.9582690.958269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ115(102,⋅)\chi_{115} (102, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :1/2), −0.951+0.307i)(2,\ 115,\ (\ :1/2),\ -0.951 + 0.307i)

Particular Values

L(1)L(1) ≈\approx 0.143611−0.910034i0.143611 - 0.910034i
L(12)L(\frac12) ≈\approx 0.143611−0.910034i0.143611 - 0.910034i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(−0.343+2.20i)T 1 + (-0.343 + 2.20i)T
23 1+(−4.64−1.20i)T 1 + (-4.64 - 1.20i)T
good2 1+(−0.174+2.43i)T+(−1.97−0.284i)T2 1 + (-0.174 + 2.43i)T + (-1.97 - 0.284i)T^{2}
3 1+(0.591−0.128i)T+(2.72−1.24i)T2 1 + (0.591 - 0.128i)T + (2.72 - 1.24i)T^{2}
7 1+(−0.0839+0.0458i)T+(3.78−5.88i)T2 1 + (-0.0839 + 0.0458i)T + (3.78 - 5.88i)T^{2}
11 1+(−2.37−2.06i)T+(1.56+10.8i)T2 1 + (-2.37 - 2.06i)T + (1.56 + 10.8i)T^{2}
13 1+(−3.02+5.54i)T+(−7.02−10.9i)T2 1 + (-3.02 + 5.54i)T + (-7.02 - 10.9i)T^{2}
17 1+(−3.26−2.44i)T+(4.78+16.3i)T2 1 + (-3.26 - 2.44i)T + (4.78 + 16.3i)T^{2}
19 1+(−0.180+1.25i)T+(−18.2−5.35i)T2 1 + (-0.180 + 1.25i)T + (-18.2 - 5.35i)T^{2}
29 1+(−2.05+0.295i)T+(27.8−8.17i)T2 1 + (-2.05 + 0.295i)T + (27.8 - 8.17i)T^{2}
31 1+(0.614−0.395i)T+(12.8−28.1i)T2 1 + (0.614 - 0.395i)T + (12.8 - 28.1i)T^{2}
37 1+(3.96+1.47i)T+(27.9+24.2i)T2 1 + (3.96 + 1.47i)T + (27.9 + 24.2i)T^{2}
41 1+(2.69−5.90i)T+(−26.8−30.9i)T2 1 + (2.69 - 5.90i)T + (-26.8 - 30.9i)T^{2}
43 1+(−1.53−7.06i)T+(−39.1+17.8i)T2 1 + (-1.53 - 7.06i)T + (-39.1 + 17.8i)T^{2}
47 1+(−3.91−3.91i)T+47iT2 1 + (-3.91 - 3.91i)T + 47iT^{2}
53 1+(5.23+9.58i)T+(−28.6+44.5i)T2 1 + (5.23 + 9.58i)T + (-28.6 + 44.5i)T^{2}
59 1+(−4.23−14.4i)T+(−49.6+31.8i)T2 1 + (-4.23 - 14.4i)T + (-49.6 + 31.8i)T^{2}
61 1+(−1.76−2.74i)T+(−25.3+55.4i)T2 1 + (-1.76 - 2.74i)T + (-25.3 + 55.4i)T^{2}
67 1+(5.91+0.422i)T+(66.3+9.53i)T2 1 + (5.91 + 0.422i)T + (66.3 + 9.53i)T^{2}
71 1+(4.76+5.49i)T+(−10.1+70.2i)T2 1 + (4.76 + 5.49i)T + (-10.1 + 70.2i)T^{2}
73 1+(1.29+1.73i)T+(−20.5+70.0i)T2 1 + (1.29 + 1.73i)T + (-20.5 + 70.0i)T^{2}
79 1+(−10.8+3.18i)T+(66.4−42.7i)T2 1 + (-10.8 + 3.18i)T + (66.4 - 42.7i)T^{2}
83 1+(−1.69+4.53i)T+(−62.7−54.3i)T2 1 + (-1.69 + 4.53i)T + (-62.7 - 54.3i)T^{2}
89 1+(6.50+4.17i)T+(36.9+80.9i)T2 1 + (6.50 + 4.17i)T + (36.9 + 80.9i)T^{2}
97 1+(2.58+6.94i)T+(−73.3+63.5i)T2 1 + (2.58 + 6.94i)T + (-73.3 + 63.5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.81821711956324289123769983407, −12.06384727294051108982712656338, −11.11066878464696587108576820759, −10.23134329691274936027008263453, −9.154691722635157369786352027335, −8.176822300205084186690970891764, −5.73867004085767892960783955568, −4.65492235634227742872557087347, −3.17105848088928327037185957415, −1.21904949803798579639881796378, 3.66693221048591655252416168619, 5.43630510890021560520115243476, 6.44215598729489501532022461946, 6.99207114366562511596914615846, 8.492050539517107373499475997102, 9.328241871343850709396531538443, 11.01349227118960675134098693831, 11.95382992010966191460249040251, 13.83628017356076374416703392280, 14.08263413973384348312851398498

Graph of the ZZ-function along the critical line