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Properties

Label 2-2695-2695.1264-c0-0-2
Degree $2$
Conductor $2695$
Sign $0.718 - 0.695i$
Analytic cond. $1.34498$
Root an. cond. $1.15973$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.88 + 0.284i)2-s + (2.53 + 0.781i)4-s + (0.0747 + 0.997i)5-s + (−0.623 − 0.781i)7-s + (2.84 + 1.37i)8-s + (0.365 − 0.930i)9-s + (−0.142 + 1.90i)10-s + (0.365 + 0.930i)11-s + (−0.455 + 0.571i)13-s + (−0.955 − 1.65i)14-s + (2.79 + 1.90i)16-s + (−1.32 − 1.22i)17-s + (0.955 − 1.65i)18-s + (−0.590 + 2.58i)20-s + (0.425 + 1.86i)22-s + ⋯
L(s)  = 1  + (1.88 + 0.284i)2-s + (2.53 + 0.781i)4-s + (0.0747 + 0.997i)5-s + (−0.623 − 0.781i)7-s + (2.84 + 1.37i)8-s + (0.365 − 0.930i)9-s + (−0.142 + 1.90i)10-s + (0.365 + 0.930i)11-s + (−0.455 + 0.571i)13-s + (−0.955 − 1.65i)14-s + (2.79 + 1.90i)16-s + (−1.32 − 1.22i)17-s + (0.955 − 1.65i)18-s + (−0.590 + 2.58i)20-s + (0.425 + 1.86i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $0.718 - 0.695i$
Analytic conductor: \(1.34498\)
Root analytic conductor: \(1.15973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2695} (1264, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2695,\ (\ :0),\ 0.718 - 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.717719503\)
\(L(\frac12)\) \(\approx\) \(3.717719503\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.0747 - 0.997i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (-0.365 - 0.930i)T \)
good2 \( 1 + (-1.88 - 0.284i)T + (0.955 + 0.294i)T^{2} \)
3 \( 1 + (-0.365 + 0.930i)T^{2} \)
13 \( 1 + (0.455 - 0.571i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (1.32 + 1.22i)T + (0.0747 + 0.997i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.826 + 0.563i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (1.78 - 0.858i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.955 - 0.294i)T^{2} \)
53 \( 1 + (-0.826 - 0.563i)T^{2} \)
59 \( 1 + (-0.0546 + 0.728i)T + (-0.988 - 0.149i)T^{2} \)
61 \( 1 + (-0.826 + 0.563i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.425 + 1.86i)T + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (-1.63 + 0.246i)T + (0.955 - 0.294i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.722 - 1.84i)T + (-0.733 - 0.680i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.424032637345551973859517560885, −7.74757345638506957335557577767, −7.08333171457563283089778138660, −6.52522053441582027905092448905, −6.44520549275085876620014555915, −4.98839888120436212650780441956, −4.32736100716790466135109822015, −3.68952737293232065767468327150, −2.87021352539814799554379538969, −1.99561542600636553158483207716, 1.63126418458918919616004466791, 2.45171403427100903590569369929, 3.43203423268548854165984110148, 4.26690701340183357443775992708, 5.02529617027298167675602694040, 5.59221013494507281117986730606, 6.27483089073410796940817337262, 7.00153926577918254218642432809, 8.208946612744423351956174304086, 8.759190690588576113204841214304

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