Properties

Label 2-845-169.118-c1-0-21
Degree $2$
Conductor $845$
Sign $0.975 - 0.219i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.466 − 0.413i)2-s + (−0.345 + 0.911i)3-s + (−0.194 − 1.60i)4-s + (0.970 + 0.239i)5-s + (0.537 − 0.282i)6-s + (−0.0247 + 0.0359i)7-s + (−1.27 + 1.85i)8-s + (1.53 + 1.35i)9-s + (−0.353 − 0.512i)10-s + (0.645 − 0.572i)11-s + (1.52 + 0.375i)12-s + (3.60 + 0.102i)13-s + (0.0263 − 0.00650i)14-s + (−0.553 + 0.801i)15-s + (−1.76 + 0.436i)16-s + (−3.15 + 4.57i)17-s + ⋯
L(s)  = 1  + (−0.329 − 0.292i)2-s + (−0.199 + 0.525i)3-s + (−0.0971 − 0.800i)4-s + (0.434 + 0.107i)5-s + (0.219 − 0.115i)6-s + (−0.00937 + 0.0135i)7-s + (−0.451 + 0.654i)8-s + (0.511 + 0.453i)9-s + (−0.111 − 0.162i)10-s + (0.194 − 0.172i)11-s + (0.440 + 0.108i)12-s + (0.999 + 0.0285i)13-s + (0.00705 − 0.00173i)14-s + (−0.142 + 0.207i)15-s + (−0.442 + 0.109i)16-s + (−0.765 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.975 - 0.219i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (456, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.975 - 0.219i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31850 + 0.146788i\)
\(L(\frac12)\) \(\approx\) \(1.31850 + 0.146788i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.970 - 0.239i)T \)
13 \( 1 + (-3.60 - 0.102i)T \)
good2 \( 1 + (0.466 + 0.413i)T + (0.241 + 1.98i)T^{2} \)
3 \( 1 + (0.345 - 0.911i)T + (-2.24 - 1.98i)T^{2} \)
7 \( 1 + (0.0247 - 0.0359i)T + (-2.48 - 6.54i)T^{2} \)
11 \( 1 + (-0.645 + 0.572i)T + (1.32 - 10.9i)T^{2} \)
17 \( 1 + (3.15 - 4.57i)T + (-6.02 - 15.8i)T^{2} \)
19 \( 1 + 2.45T + 19T^{2} \)
23 \( 1 - 4.88T + 23T^{2} \)
29 \( 1 + (-0.577 - 0.511i)T + (3.49 + 28.7i)T^{2} \)
31 \( 1 + (-2.50 + 1.31i)T + (17.6 - 25.5i)T^{2} \)
37 \( 1 + (-8.93 + 4.68i)T + (21.0 - 30.4i)T^{2} \)
41 \( 1 + (-1.22 + 3.23i)T + (-30.6 - 27.1i)T^{2} \)
43 \( 1 + (-4.97 - 2.61i)T + (24.4 + 35.3i)T^{2} \)
47 \( 1 + (1.09 - 9.05i)T + (-45.6 - 11.2i)T^{2} \)
53 \( 1 + (-1.90 + 2.76i)T + (-18.7 - 49.5i)T^{2} \)
59 \( 1 + (-1.97 - 0.487i)T + (52.2 + 27.4i)T^{2} \)
61 \( 1 + (6.02 + 8.72i)T + (-21.6 + 57.0i)T^{2} \)
67 \( 1 + (1.59 - 13.1i)T + (-65.0 - 16.0i)T^{2} \)
71 \( 1 + (-0.586 + 1.54i)T + (-53.1 - 47.0i)T^{2} \)
73 \( 1 + (-9.70 + 8.59i)T + (8.79 - 72.4i)T^{2} \)
79 \( 1 + (-1.59 + 13.1i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (-3.98 - 10.5i)T + (-62.1 + 55.0i)T^{2} \)
89 \( 1 - 4.58T + 89T^{2} \)
97 \( 1 + (-0.804 + 0.198i)T + (85.8 - 45.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.42788689973702651485921086225, −9.375105452868978524924028210580, −8.932951810278256046041970389570, −7.83826427399254377051139847859, −6.43269253628708092653544964833, −5.95401433335362064767540312208, −4.84306264546500823714552446494, −3.99831666574837751367247704198, −2.39176469877581449622220902059, −1.24282022686611987946039869271, 0.904238825146340922814372717082, 2.50875806135483450910682003724, 3.76458521046221977888796641645, 4.75685358020705632641516379050, 6.20219550013116480335621892864, 6.76060354623192430227399963681, 7.49587982300174334887741794703, 8.545489249949328707958719904279, 9.138135128092646553429165698119, 9.936133877376944609337235989727

Graph of the $Z$-function along the critical line