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Properties

Label 2-2695-2695.879-c0-0-0
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  + (−0.496 + 1.26i)2-s + (−0.623 − 0.578i)4-s + (−0.826 − 0.563i)5-s + (0.781 + 0.623i)7-s + (−0.183 + 0.0882i)8-s + (−0.988 − 0.149i)9-s + (1.12 − 0.766i)10-s + (0.988 − 0.149i)11-s + (0.185 + 0.233i)13-s + (−1.17 + 0.680i)14-s + (−0.0842 − 1.12i)16-s + (0.829 + 0.255i)17-s + (0.680 − 1.17i)18-s + (0.189 + 0.829i)20-s + (−0.302 + 1.32i)22-s + ⋯
L(s)  = 1  + (−0.496 + 1.26i)2-s + (−0.623 − 0.578i)4-s + (−0.826 − 0.563i)5-s + (0.781 + 0.623i)7-s + (−0.183 + 0.0882i)8-s + (−0.988 − 0.149i)9-s + (1.12 − 0.766i)10-s + (0.988 − 0.149i)11-s + (0.185 + 0.233i)13-s + (−1.17 + 0.680i)14-s + (−0.0842 − 1.12i)16-s + (0.829 + 0.255i)17-s + (0.680 − 1.17i)18-s + (0.189 + 0.829i)20-s + (−0.302 + 1.32i)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: $\chi_{2695} (879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2695,\ (\ :0),\ -0.718 - 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7841336547\)
\(L(\frac12)\) \(\approx\) \(0.7841336547\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.826 + 0.563i)T \)
7 \( 1 + (-0.781 - 0.623i)T \)
11 \( 1 + (-0.988 + 0.149i)T \)
good2 \( 1 + (0.496 - 1.26i)T + (-0.733 - 0.680i)T^{2} \)
3 \( 1 + (0.988 + 0.149i)T^{2} \)
13 \( 1 + (-0.185 - 0.233i)T + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + (-0.829 - 0.255i)T + (0.826 + 0.563i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.826 + 0.563i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.0747 + 0.997i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-1.67 - 0.807i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.733 + 0.680i)T^{2} \)
53 \( 1 + (-0.0747 - 0.997i)T^{2} \)
59 \( 1 + (1.63 - 1.11i)T + (0.365 - 0.930i)T^{2} \)
61 \( 1 + (-0.0747 + 0.997i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.326 + 1.42i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.728 - 1.85i)T + (-0.733 + 0.680i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.07 - 1.35i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.722 - 0.108i)T + (0.955 + 0.294i)T^{2} \)
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

−9.015876718739275069251284408156, −8.420169480782557111754960258682, −7.923191160619563044525493804799, −7.20826901231741039267182770211, −6.23754318165434744791557249853, −5.64890072029216468835447608492, −4.92376732702791931622739943492, −3.89060894563345141035451981089, −2.83925595087060949420068231606, −1.22416857843272225039711025874, 0.69857111474213093599801790275, 1.90597792739310737185477140372, 2.96763787266837347097950731164, 3.67632825674986544835017254663, 4.37780987389798907681202849478, 5.62203148522713636132939400721, 6.52920568139613575052875683193, 7.52039538294681336767129692881, 8.071765596379128686187035036100, 8.895108949436773114714943859409

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