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Properties

Label 2-795-795.317-c1-0-2
Degree $2$
Conductor $795$
Sign $-0.108 - 0.994i$
Analytic cond. $6.34810$
Root an. cond. $2.51954$
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  + (−0.278 − 0.278i)2-s + (−1.22 + 1.22i)3-s − 1.84i·4-s + (−2.01 − 0.962i)5-s + 0.682·6-s + (−1.49 − 1.49i)7-s + (−1.07 + 1.07i)8-s − 2.99i·9-s + (0.294 + 0.830i)10-s + (2.25 + 2.25i)12-s + (0.00506 − 0.00506i)13-s + 0.835i·14-s + (3.65 − 1.29i)15-s − 3.09·16-s + (−0.836 + 0.836i)18-s + ⋯
L(s)  = 1  + (−0.197 − 0.197i)2-s + (−0.707 + 0.707i)3-s − 0.922i·4-s + (−0.902 − 0.430i)5-s + 0.278·6-s + (−0.566 − 0.566i)7-s + (−0.378 + 0.378i)8-s − 0.999i·9-s + (0.0930 + 0.262i)10-s + (0.652 + 0.652i)12-s + (0.00140 − 0.00140i)13-s + 0.223i·14-s + (0.942 − 0.333i)15-s − 0.773·16-s + (−0.197 + 0.197i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 795 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 795 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(795\)    =    \(3 \cdot 5 \cdot 53\)
Sign: $-0.108 - 0.994i$
Analytic conductor: \(6.34810\)
Root analytic conductor: \(2.51954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{795} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 795,\ (\ :1/2),\ -0.108 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140037 + 0.156112i\)
\(L(\frac12)\) \(\approx\) \(0.140037 + 0.156112i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + (2.01 + 0.962i)T \)
53 \( 1 + (5.14 - 5.14i)T \)
good2 \( 1 + (0.278 + 0.278i)T + 2iT^{2} \)
7 \( 1 + (1.49 + 1.49i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-0.00506 + 0.00506i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (1.59 - 1.59i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-7.59 - 7.59i)T + 37iT^{2} \)
41 \( 1 + 1.59T + 41T^{2} \)
43 \( 1 + (7.83 - 7.83i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 + 4.78T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (4.14 - 4.14i)T - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-9.81 - 9.81i)T + 97iT^{2} \)
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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

−10.41865439412512201379403616794, −9.820686947599303199365511274244, −9.111668339856831577717617485357, −8.053583320024904019442058778290, −6.83970117657803737074204931016, −6.07555487048111716483997722167, −5.01768998332113809243367783157, −4.30987132486385997711625668063, −3.21797630924427101119563674481, −1.11427584964696830973785804932, 0.14328974754784996774032678538, 2.39551697104165525366872828685, 3.45705691863749189267562397006, 4.57823456651408363703455441150, 5.92844847115777830182388889041, 6.72852357702372861025528205792, 7.42817742348426093283508088187, 8.140109741094699727508679108204, 8.934974351442903701934978688444, 10.13592331651955751871013430347

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