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Properties

Label 2-2695-385.219-c0-0-1
Degree $2$
Conductor $2695$
Sign $-0.701 - 0.712i$
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  + (0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)16-s − 0.999·20-s + (−0.499 − 0.866i)25-s + (−1 − 1.73i)31-s − 0.999·36-s + (−0.499 + 0.866i)44-s + (−0.499 − 0.866i)45-s − 0.999·55-s + (1 + 1.73i)59-s − 0.999·64-s + 2·71-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)16-s − 0.999·20-s + (−0.499 − 0.866i)25-s + (−1 − 1.73i)31-s − 0.999·36-s + (−0.499 + 0.866i)44-s + (−0.499 − 0.866i)45-s − 0.999·55-s + (1 + 1.73i)59-s − 0.999·64-s + 2·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(1.34498\)
Root analytic conductor: \(1.15973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2695} (2529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2695,\ (\ :0),\ -0.701 - 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.068434491\)
\(L(\frac12)\) \(\approx\) \(1.068434491\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.244611787527079797980879764905, −8.296801039681925840484531225610, −7.71016037166296572643798646451, −7.17033357894042585911673004590, −6.47971998146522973978615674705, −5.52134702214018870609160057552, −4.31641117516644390309647506704, −3.72401527630419638520141963210, −2.66337139675600812422663668663, −2.06115973498058652356918973525, 0.67267570652625059097701596743, 1.68466585661860629226360305286, 3.11765446951708870150451453207, 3.87414775661912654685583023498, 5.03468148405383673387070039448, 5.59798230067482448415064439656, 6.43019035276263882179334810122, 7.05603875725424605697304089906, 8.148495611327979792979006574257, 8.840965717356467102304197631883

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