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Properties

Label 2-261-261.101-c1-0-5
Degree 22
Conductor 261261
Sign −0.704−0.709i-0.704 - 0.709i
Analytic cond. 2.084092.08409
Root an. cond. 1.443631.44363
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  + (−0.942 + 2.16i)2-s + (−1.73 + 0.0507i)3-s + (−2.41 − 2.60i)4-s + (3.53 + 0.532i)5-s + (1.52 − 3.78i)6-s + (2.36 + 2.19i)7-s + (3.45 − 1.20i)8-s + (2.99 − 0.175i)9-s + (−4.48 + 7.13i)10-s + (4.16 − 3.58i)11-s + (4.31 + 4.38i)12-s + (−3.02 + 4.43i)13-s + (−6.96 + 3.04i)14-s + (−6.14 − 0.743i)15-s + (−0.113 + 1.52i)16-s + (0.320 + 0.320i)17-s + ⋯
L(s)  = 1  + (−0.666 + 1.52i)2-s + (−0.999 + 0.0292i)3-s + (−1.20 − 1.30i)4-s + (1.58 + 0.238i)5-s + (0.621 − 1.54i)6-s + (0.893 + 0.829i)7-s + (1.22 − 0.427i)8-s + (0.998 − 0.0585i)9-s + (−1.41 + 2.25i)10-s + (1.25 − 1.08i)11-s + (1.24 + 1.26i)12-s + (−0.838 + 1.23i)13-s + (−1.86 + 0.812i)14-s + (−1.58 − 0.191i)15-s + (−0.0284 + 0.380i)16-s + (0.0776 + 0.0776i)17-s + ⋯

Functional equation

Λ(s)=(261s/2 ΓC(s) L(s)=((−0.704−0.709i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(261s/2 ΓC(s+1/2) L(s)=((−0.704−0.709i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 261261    =    32⋅293^{2} \cdot 29
Sign: −0.704−0.709i-0.704 - 0.709i
Analytic conductor: 2.084092.08409
Root analytic conductor: 1.443631.44363
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ261(101,⋅)\chi_{261} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 261, ( :1/2), −0.704−0.709i)(2,\ 261,\ (\ :1/2),\ -0.704 - 0.709i)

Particular Values

L(1)L(1) ≈\approx 0.343041+0.824375i0.343041 + 0.824375i
L(12)L(\frac12) ≈\approx 0.343041+0.824375i0.343041 + 0.824375i
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)
bad3 1+(1.73−0.0507i)T 1 + (1.73 - 0.0507i)T
29 1+(−3.80−3.80i)T 1 + (-3.80 - 3.80i)T
good2 1+(0.942−2.16i)T+(−1.36−1.46i)T2 1 + (0.942 - 2.16i)T + (-1.36 - 1.46i)T^{2}
5 1+(−3.53−0.532i)T+(4.77+1.47i)T2 1 + (-3.53 - 0.532i)T + (4.77 + 1.47i)T^{2}
7 1+(−2.36−2.19i)T+(0.523+6.98i)T2 1 + (-2.36 - 2.19i)T + (0.523 + 6.98i)T^{2}
11 1+(−4.16+3.58i)T+(1.63−10.8i)T2 1 + (-4.16 + 3.58i)T + (1.63 - 10.8i)T^{2}
13 1+(3.02−4.43i)T+(−4.74−12.1i)T2 1 + (3.02 - 4.43i)T + (-4.74 - 12.1i)T^{2}
17 1+(−0.320−0.320i)T+17iT2 1 + (-0.320 - 0.320i)T + 17iT^{2}
19 1+(1.53+0.962i)T+(8.24+17.1i)T2 1 + (1.53 + 0.962i)T + (8.24 + 17.1i)T^{2}
23 1+(2.14+0.843i)T+(16.8+15.6i)T2 1 + (2.14 + 0.843i)T + (16.8 + 15.6i)T^{2}
31 1+(3.49−2.57i)T+(9.13−29.6i)T2 1 + (3.49 - 2.57i)T + (9.13 - 29.6i)T^{2}
37 1+(1.06+3.04i)T+(−28.9+23.0i)T2 1 + (1.06 + 3.04i)T + (-28.9 + 23.0i)T^{2}
41 1+(−0.766−0.205i)T+(35.5+20.5i)T2 1 + (-0.766 - 0.205i)T + (35.5 + 20.5i)T^{2}
43 1+(3.30−4.48i)T+(−12.6−41.0i)T2 1 + (3.30 - 4.48i)T + (-12.6 - 41.0i)T^{2}
47 1+(6.95+8.08i)T+(−7.00+46.4i)T2 1 + (6.95 + 8.08i)T + (-7.00 + 46.4i)T^{2}
53 1+(−5.65−4.50i)T+(11.7+51.6i)T2 1 + (-5.65 - 4.50i)T + (11.7 + 51.6i)T^{2}
59 1+(2.18−1.26i)T+(29.5−51.0i)T2 1 + (2.18 - 1.26i)T + (29.5 - 51.0i)T^{2}
61 1+(−3.14+0.117i)T+(60.8−4.55i)T2 1 + (-3.14 + 0.117i)T + (60.8 - 4.55i)T^{2}
67 1+(3.09−0.232i)T+(66.2−9.98i)T2 1 + (3.09 - 0.232i)T + (66.2 - 9.98i)T^{2}
71 1+(−6.93+3.33i)T+(44.2−55.5i)T2 1 + (-6.93 + 3.33i)T + (44.2 - 55.5i)T^{2}
73 1+(0.346−3.07i)T+(−71.1−16.2i)T2 1 + (0.346 - 3.07i)T + (-71.1 - 16.2i)T^{2}
79 1+(−1.51+7.99i)T+(−73.5−28.8i)T2 1 + (-1.51 + 7.99i)T + (-73.5 - 28.8i)T^{2}
83 1+(−4.05+13.1i)T+(−68.5−46.7i)T2 1 + (-4.05 + 13.1i)T + (-68.5 - 46.7i)T^{2}
89 1+(6.94−0.782i)T+(86.7−19.8i)T2 1 + (6.94 - 0.782i)T + (86.7 - 19.8i)T^{2}
97 1+(3.86+7.30i)T+(−54.6+80.1i)T2 1 + (3.86 + 7.30i)T + (-54.6 + 80.1i)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

−12.18760818946629453825574446925, −11.29880971825678455161262558330, −10.10042134544011174001540661031, −9.228619310074496963501413001233, −8.620715703758396228637305939979, −6.99997174037724354628355013101, −6.35327367603866034478042385343, −5.66001903008155474325797244820, −4.80283542580538541282275430375, −1.70582819848014279026068156788, 1.16536544184547951963344689542, 2.08878574475028847823387699995, 4.18101902599853369480873287184, 5.22276017073838258683312448932, 6.57295461610599737290345041268, 7.920128392127259030175325991719, 9.435993041135379701184633824491, 9.988810984706895561963064395492, 10.50231632113624699752957615053, 11.51232868135763234485841986771

Graph of the ZZ-function along the critical line