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Properties

Label 2-845-13.12-c1-0-38
Degree 22
Conductor 845845
Sign −0.554+0.832i-0.554 + 0.832i
Analytic cond. 6.747356.74735
Root an. cond. 2.597562.59756
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  − 1.30i·2-s + 3-s + 0.302·4-s + i·5-s − 1.30i·6-s − i·7-s − 3i·8-s − 2·9-s + 1.30·10-s − 5.60i·11-s + 0.302·12-s − 1.30·14-s + i·15-s − 3.30·16-s − 0.394·17-s + 2.60i·18-s + ⋯
L(s)  = 1  − 0.921i·2-s + 0.577·3-s + 0.151·4-s + 0.447i·5-s − 0.531i·6-s − 0.377i·7-s − 1.06i·8-s − 0.666·9-s + 0.411·10-s − 1.69i·11-s + 0.0874·12-s − 0.348·14-s + 0.258i·15-s − 0.825·16-s − 0.0956·17-s + 0.614i·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((−0.554+0.832i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((−0.554+0.832i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: −0.554+0.832i-0.554 + 0.832i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(506,⋅)\chi_{845} (506, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), −0.554+0.832i)(2,\ 845,\ (\ :1/2),\ -0.554 + 0.832i)

Particular Values

L(1)L(1) ≈\approx 0.900838−1.68323i0.900838 - 1.68323i
L(12)L(\frac12) ≈\approx 0.900838−1.68323i0.900838 - 1.68323i
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−iT 1 - iT
13 1 1
good2 1+1.30iT−2T2 1 + 1.30iT - 2T^{2}
3 1−T+3T2 1 - T + 3T^{2}
7 1+iT−7T2 1 + iT - 7T^{2}
11 1+5.60iT−11T2 1 + 5.60iT - 11T^{2}
17 1+0.394T+17T2 1 + 0.394T + 17T^{2}
19 1+1.60iT−19T2 1 + 1.60iT - 19T^{2}
23 1−3T+23T2 1 - 3T + 23T^{2}
29 1−8.21T+29T2 1 - 8.21T + 29T^{2}
31 1−4iT−31T2 1 - 4iT - 31T^{2}
37 1+3.60iT−37T2 1 + 3.60iT - 37T^{2}
41 1+3iT−41T2 1 + 3iT - 41T^{2}
43 1+4.21T+43T2 1 + 4.21T + 43T^{2}
47 1+5.21iT−47T2 1 + 5.21iT - 47T^{2}
53 1−11.2T+53T2 1 - 11.2T + 53T^{2}
59 1−10.8iT−59T2 1 - 10.8iT - 59T^{2}
61 1+T+61T2 1 + T + 61T^{2}
67 1−7iT−67T2 1 - 7iT - 67T^{2}
71 1−16.8iT−71T2 1 - 16.8iT - 71T^{2}
73 1+15.2iT−73T2 1 + 15.2iT - 73T^{2}
79 1+9.21T+79T2 1 + 9.21T + 79T^{2}
83 1+5.21iT−83T2 1 + 5.21iT - 83T^{2}
89 1−8.21iT−89T2 1 - 8.21iT - 89T^{2}
97 1−15.6iT−97T2 1 - 15.6iT - 97T^{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

−10.27054069105437725934949483021, −8.980216828612454222552741662373, −8.522671620932501222916620042011, −7.33783845192385672678610007151, −6.50498729328812230102940576659, −5.52035299801190722034314116864, −3.93929568582082845433700094046, −3.12614132770711678065679383745, −2.51561191865469427255836164011, −0.859888855228293190432234916455, 1.93123094981391850820771288660, 2.87673393773674537621593673624, 4.46615198507554606612259603737, 5.27961296603731997183166957700, 6.25447385130200532533504173511, 7.11750543823000065923678209129, 7.949827979854183553008647373506, 8.569863572290455158754656281074, 9.390240678583582840597506413423, 10.26849664446102584539420330658

Graph of the ZZ-function along the critical line