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Properties

Label 2-845-65.32-c1-0-43
Degree 22
Conductor 845845
Sign −0.333+0.942i-0.333 + 0.942i
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  + (−0.246 + 0.427i)2-s + (−0.243 − 0.908i)3-s + (0.878 + 1.52i)4-s + (−2.21 + 0.284i)5-s + (0.448 + 0.120i)6-s + (−3.18 + 1.83i)7-s − 1.85·8-s + (1.83 − 1.05i)9-s + (0.426 − 1.01i)10-s + (0.664 − 0.177i)11-s + (1.16 − 1.16i)12-s − 1.81i·14-s + (0.798 + 1.94i)15-s + (−1.29 + 2.24i)16-s + (−2.29 − 0.614i)17-s + 1.04i·18-s + ⋯
L(s)  = 1  + (−0.174 + 0.302i)2-s + (−0.140 − 0.524i)3-s + (0.439 + 0.760i)4-s + (−0.991 + 0.127i)5-s + (0.183 + 0.0490i)6-s + (−1.20 + 0.694i)7-s − 0.655·8-s + (0.610 − 0.352i)9-s + (0.134 − 0.322i)10-s + (0.200 − 0.0536i)11-s + (0.337 − 0.337i)12-s − 0.485i·14-s + (0.206 + 0.502i)15-s + (−0.324 + 0.562i)16-s + (−0.556 − 0.149i)17-s + 0.246i·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((−0.333+0.942i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((−0.333+0.942i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: −0.333+0.942i-0.333 + 0.942i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(357,⋅)\chi_{845} (357, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), −0.333+0.942i)(2,\ 845,\ (\ :1/2),\ -0.333 + 0.942i)

Particular Values

L(1)L(1) ≈\approx 0.192169−0.271900i0.192169 - 0.271900i
L(12)L(\frac12) ≈\approx 0.192169−0.271900i0.192169 - 0.271900i
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+(2.21−0.284i)T 1 + (2.21 - 0.284i)T
13 1 1
good2 1+(0.246−0.427i)T+(−1−1.73i)T2 1 + (0.246 - 0.427i)T + (-1 - 1.73i)T^{2}
3 1+(0.243+0.908i)T+(−2.59+1.5i)T2 1 + (0.243 + 0.908i)T + (-2.59 + 1.5i)T^{2}
7 1+(3.18−1.83i)T+(3.5−6.06i)T2 1 + (3.18 - 1.83i)T + (3.5 - 6.06i)T^{2}
11 1+(−0.664+0.177i)T+(9.52−5.5i)T2 1 + (-0.664 + 0.177i)T + (9.52 - 5.5i)T^{2}
17 1+(2.29+0.614i)T+(14.7+8.5i)T2 1 + (2.29 + 0.614i)T + (14.7 + 8.5i)T^{2}
19 1+(−1.41+5.29i)T+(−16.4−9.5i)T2 1 + (-1.41 + 5.29i)T + (-16.4 - 9.5i)T^{2}
23 1+(1.30−0.350i)T+(19.9−11.5i)T2 1 + (1.30 - 0.350i)T + (19.9 - 11.5i)T^{2}
29 1+(8.24+4.75i)T+(14.5+25.1i)T2 1 + (8.24 + 4.75i)T + (14.5 + 25.1i)T^{2}
31 1+(−4.81+4.81i)T−31iT2 1 + (-4.81 + 4.81i)T - 31iT^{2}
37 1+(1.58+0.917i)T+(18.5+32.0i)T2 1 + (1.58 + 0.917i)T + (18.5 + 32.0i)T^{2}
41 1+(−0.143−0.534i)T+(−35.5+20.5i)T2 1 + (-0.143 - 0.534i)T + (-35.5 + 20.5i)T^{2}
43 1+(−0.560+2.09i)T+(−37.2−21.5i)T2 1 + (-0.560 + 2.09i)T + (-37.2 - 21.5i)T^{2}
47 1+3.80iT−47T2 1 + 3.80iT - 47T^{2}
53 1+(2.47−2.47i)T−53iT2 1 + (2.47 - 2.47i)T - 53iT^{2}
59 1+(10.0+2.69i)T+(51.0+29.5i)T2 1 + (10.0 + 2.69i)T + (51.0 + 29.5i)T^{2}
61 1+(3.09+5.36i)T+(−30.5+52.8i)T2 1 + (3.09 + 5.36i)T + (-30.5 + 52.8i)T^{2}
67 1+(6.12−10.6i)T+(−33.5−58.0i)T2 1 + (6.12 - 10.6i)T + (-33.5 - 58.0i)T^{2}
71 1+(−6.47−1.73i)T+(61.4+35.5i)T2 1 + (-6.47 - 1.73i)T + (61.4 + 35.5i)T^{2}
73 1−3.37T+73T2 1 - 3.37T + 73T^{2}
79 1−3.12iT−79T2 1 - 3.12iT - 79T^{2}
83 1+2.13iT−83T2 1 + 2.13iT - 83T^{2}
89 1+(−0.874−3.26i)T+(−77.0+44.5i)T2 1 + (-0.874 - 3.26i)T + (-77.0 + 44.5i)T^{2}
97 1+(3.53+6.12i)T+(−48.5+84.0i)T2 1 + (3.53 + 6.12i)T + (-48.5 + 84.0i)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

−9.640840637464054994572142916598, −9.052831113282643803220244133445, −8.050145642323936342704445287019, −7.25313428617894357153381029036, −6.70798643524016893761895783413, −5.96034902034883923335286917610, −4.29141484449157275650790272621, −3.39294350653949704448134200566, −2.41671568761304740805151693932, −0.17421643556788183511488700842, 1.46160846735637784424877193545, 3.20929261794699247055866733577, 3.98849099222660148341990291921, 5.01527158154313386030751701369, 6.20811918634472345703347149208, 6.99697699062574340610920009672, 7.78672505957026000775408970782, 9.112624090862479994726863979768, 9.751298061257776683999260502890, 10.56620972087949099081357707931

Graph of the ZZ-function along the critical line