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Properties

Label 2-2695-2695.1649-c0-0-0
Degree $2$
Conductor $2695$
Sign $-0.972 - 0.232i$
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  + (−0.147 − 1.97i)2-s + (−2.87 + 0.433i)4-s + (−0.733 + 0.680i)5-s + (−0.900 + 0.433i)7-s + (0.841 + 3.68i)8-s + (0.826 + 0.563i)9-s + (1.44 + 1.34i)10-s + (0.826 − 0.563i)11-s + (−1.48 − 0.716i)13-s + (0.988 + 1.71i)14-s + (4.36 − 1.34i)16-s + (−0.162 − 0.414i)17-s + (0.988 − 1.71i)18-s + (1.81 − 2.27i)20-s + (−1.23 − 1.54i)22-s + ⋯
L(s)  = 1  + (−0.147 − 1.97i)2-s + (−2.87 + 0.433i)4-s + (−0.733 + 0.680i)5-s + (−0.900 + 0.433i)7-s + (0.841 + 3.68i)8-s + (0.826 + 0.563i)9-s + (1.44 + 1.34i)10-s + (0.826 − 0.563i)11-s + (−1.48 − 0.716i)13-s + (0.988 + 1.71i)14-s + (4.36 − 1.34i)16-s + (−0.162 − 0.414i)17-s + (0.988 − 1.71i)18-s + (1.81 − 2.27i)20-s + (−1.23 − 1.54i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $-0.972 - 0.232i$
Analytic conductor: \(1.34498\)
Root analytic conductor: \(1.15973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2695} (1649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2695,\ (\ :0),\ -0.972 - 0.232i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4920852504\)
\(L(\frac12)\) \(\approx\) \(0.4920852504\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.733 - 0.680i)T \)
7 \( 1 + (0.900 - 0.433i)T \)
11 \( 1 + (-0.826 + 0.563i)T \)
good2 \( 1 + (0.147 + 1.97i)T + (-0.988 + 0.149i)T^{2} \)
3 \( 1 + (-0.826 - 0.563i)T^{2} \)
13 \( 1 + (1.48 + 0.716i)T + (0.623 + 0.781i)T^{2} \)
17 \( 1 + (0.162 + 0.414i)T + (-0.733 + 0.680i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.733 + 0.680i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.955 - 0.294i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.0332 - 0.145i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.988 - 0.149i)T^{2} \)
53 \( 1 + (-0.955 + 0.294i)T^{2} \)
59 \( 1 + (1.21 + 1.12i)T + (0.0747 + 0.997i)T^{2} \)
61 \( 1 + (-0.955 - 0.294i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (1.23 + 1.54i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.142 + 1.90i)T + (-0.988 - 0.149i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.900 + 0.433i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.123 - 0.0841i)T + (0.365 + 0.930i)T^{2} \)
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

−9.045609750445057255466904056179, −7.999394188313837331973243254214, −7.44802792055704216702156731220, −6.23759725052968443967798785808, −5.00757818747293625639934973915, −4.34582584295565405906368454952, −3.41970520245272223437057371787, −2.86989917044289946986208211331, −2.01400885439452585163364512414, −0.40728778331936876951914854232, 1.13411094084994508549246265985, 3.58485084488143939775264049597, 4.37572848012329689026584300911, 4.63547219861261467961533953481, 5.79529468553278497755368008840, 6.78286995904782219202965349288, 7.01540680618644955146335891347, 7.60643149833483143572621683215, 8.599992419272782682275751805787, 9.228439697233635975401487620532

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