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Properties

Label 2-2695-385.362-c0-0-4
Degree $2$
Conductor $2695$
Sign $0.343 + 0.939i$
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  + (0.198 − 0.739i)3-s + (0.866 − 0.5i)4-s + (−0.793 + 0.608i)5-s + (0.358 + 0.207i)9-s + (−0.5 − 0.866i)11-s + (−0.198 − 0.739i)12-s + (0.292 + 0.707i)15-s + (0.499 − 0.866i)16-s + (−0.382 + 0.923i)20-s + (0.258 − 0.965i)25-s + (0.765 − 0.765i)27-s + (0.662 − 0.382i)31-s + (−0.739 + 0.198i)33-s + 0.414·36-s + (1.36 − 0.366i)37-s + ⋯
L(s)  = 1  + (0.198 − 0.739i)3-s + (0.866 − 0.5i)4-s + (−0.793 + 0.608i)5-s + (0.358 + 0.207i)9-s + (−0.5 − 0.866i)11-s + (−0.198 − 0.739i)12-s + (0.292 + 0.707i)15-s + (0.499 − 0.866i)16-s + (−0.382 + 0.923i)20-s + (0.258 − 0.965i)25-s + (0.765 − 0.765i)27-s + (0.662 − 0.382i)31-s + (−0.739 + 0.198i)33-s + 0.414·36-s + (1.36 − 0.366i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $0.343 + 0.939i$
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: \((2,\ 2695,\ (\ :0),\ 0.343 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.421286999\)
\(L(\frac12)\) \(\approx\) \(1.421286999\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.793 - 0.608i)T \)
7 \( 1 \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
3 \( 1 + (-0.198 + 0.739i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.478 + 1.78i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (1.93 + 0.517i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
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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

−8.561890078456465788299154322156, −7.905117975525913958955407940817, −7.38685313040247261329375005230, −6.66467833660986059048063843395, −6.08814185348798522211075925662, −5.09012578219489210379832759324, −3.95378939788686467470480163287, −2.90013346533087166198303929544, −2.26339834877237200578393175114, −0.961659856765754162232500806501, 1.48889368413511625753574916266, 2.78643917104621330407958703797, 3.57163515681239726133557970308, 4.45072944899781495538568748209, 4.93932937549544357616864799199, 6.25245808055176328577695941285, 6.99010410592474414217473085051, 7.86953324387844841252836878230, 8.147697762182082853778918784407, 9.320494701690805151140292537067

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