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Properties

Label 2-2695-2695.879-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8216257048\)
\(L(\frac12)\) \(\approx\) \(0.8216257048\)
\(L(1)\) 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

−8.964860270099119079761280331570, −7.81904036588785143246196277056, −7.03057288377470413405201431189, −6.21873883274487994934025121201, −5.02765083704426283721644682448, −4.35568260670729902461905739711, −3.45422275558653937997918128590, −3.13587724601444518345817517650, −1.72527226069352540378099359301, −0.43095605238365912715950761446, 2.17505755125202751280839406946, 3.28645315121851422142409733969, 4.08365472845942634608247419735, 4.95809639675153839125944099297, 5.92148509237304708449967391184, 6.48319886292699417586408718129, 6.94104036052682344223065220264, 7.86830425942056680984204074167, 8.517661803075364926004682134812, 9.159689011823703352632773738558

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