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Properties

Label 2-2695-385.362-c0-0-1
Degree $2$
Conductor $2695$
Sign $0.981 + 0.193i$
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  + (−0.478 + 1.78i)3-s + (0.866 − 0.5i)4-s + (−0.608 − 0.793i)5-s + (−2.09 − 1.20i)9-s + (−0.5 − 0.866i)11-s + (0.478 + 1.78i)12-s + (1.70 − 0.707i)15-s + (0.499 − 0.866i)16-s + (−0.923 − 0.382i)20-s + (−0.258 + 0.965i)25-s + (1.84 − 1.84i)27-s + (1.60 − 0.923i)31-s + (1.78 − 0.478i)33-s − 2.41·36-s + (1.36 − 0.366i)37-s + ⋯
L(s)  = 1  + (−0.478 + 1.78i)3-s + (0.866 − 0.5i)4-s + (−0.608 − 0.793i)5-s + (−2.09 − 1.20i)9-s + (−0.5 − 0.866i)11-s + (0.478 + 1.78i)12-s + (1.70 − 0.707i)15-s + (0.499 − 0.866i)16-s + (−0.923 − 0.382i)20-s + (−0.258 + 0.965i)25-s + (1.84 − 1.84i)27-s + (1.60 − 0.923i)31-s + (1.78 − 0.478i)33-s − 2.41·36-s + (1.36 − 0.366i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $0.981 + 0.193i$
Analytic conductor: \(1.34498\)
Root analytic conductor: \(1.15973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2695} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2695,\ (\ :0),\ 0.981 + 0.193i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9853343646\)
\(L(\frac12)\) \(\approx\) \(0.9853343646\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.608 + 0.793i)T \)
7 \( 1 \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
3 \( 1 + (0.478 - 1.78i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1.60 + 0.923i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.198 + 0.739i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-1.93 - 0.517i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.517 + 1.93i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
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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

−9.176412171496895789535022576801, −8.383857491123115230811600058368, −7.66094423870591008754432117703, −6.34399948232781795776581414759, −5.74726047833942196354265905216, −5.07838373095918629139048229415, −4.36115864943185967993206552287, −3.51959532673602095208878637254, −2.61051015284575991493450757889, −0.71099184059421557461429941121, 1.31163913442791294851983040390, 2.52048874206024513591182655671, 2.80399200543493871595289629005, 4.23654492790321370837822487716, 5.51720484045657427383147223638, 6.40539288473565193777510077884, 6.84359376618588667289915563137, 7.41269350542108326146896727318, 7.953188807107627904714117357044, 8.513283486963020151795443862605

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