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Properties

Label 2-2695-385.109-c0-0-2
Degree $2$
Conductor $2695$
Sign $0.328 + 0.944i$
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  + (−1.22 − 0.707i)3-s + (0.5 − 0.866i)4-s + (−0.258 + 0.965i)5-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (−1.22 + 0.707i)12-s + (1 − 0.999i)15-s + (−0.499 − 0.866i)16-s + (0.707 + 0.707i)20-s + (1.73 − i)23-s + (−0.866 − 0.499i)25-s + (0.707 − 1.22i)31-s + (1.22 − 0.707i)33-s + 0.999·36-s + (0.499 + 0.866i)44-s + (−0.965 + 0.258i)45-s + ⋯
L(s)  = 1  + (−1.22 − 0.707i)3-s + (0.5 − 0.866i)4-s + (−0.258 + 0.965i)5-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (−1.22 + 0.707i)12-s + (1 − 0.999i)15-s + (−0.499 − 0.866i)16-s + (0.707 + 0.707i)20-s + (1.73 − i)23-s + (−0.866 − 0.499i)25-s + (0.707 − 1.22i)31-s + (1.22 − 0.707i)33-s + 0.999·36-s + (0.499 + 0.866i)44-s + (−0.965 + 0.258i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(1.34498\)
Root analytic conductor: \(1.15973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2695} (2419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2695,\ (\ :0),\ 0.328 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7829996784\)
\(L(\frac12)\) \(\approx\) \(0.7829996784\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.258 - 0.965i)T \)
7 \( 1 \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 1.41iT - T^{2} \)
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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

−8.952565841678309393584710467693, −7.64076060784873612259706004692, −7.12782232833497550880412687891, −6.61084890592465899042443091408, −5.95070040028989505877822276498, −5.22411615682836248400278104398, −4.38262605739193287721777198036, −2.84830007165578642747408836879, −2.05244296524252265900292960449, −0.71885221688529286051645531665, 1.07305421619055593239645515158, 2.77749679417656683171763146478, 3.72067288005102341631711277355, 4.54788257366943927555891010175, 5.31577927695807387920028645600, 5.85081032061875640082497510171, 6.90661779813281065210807759010, 7.60848900579968908989184195382, 8.592510303444214183176486170027, 8.947484832249715924605580129400

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