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Properties

Label 2-2925-117.70-c0-0-0
Degree $2$
Conductor $2925$
Sign $0.919 + 0.393i$
Analytic cond. $1.45976$
Root an. cond. $1.20820$
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  − 3-s + (0.866 − 0.5i)4-s + 9-s + (1.36 − 0.366i)11-s + (−0.866 + 0.5i)12-s + (0.5 + 0.866i)13-s + (0.499 − 0.866i)16-s − i·17-s + (−1 + i)19-s + (0.866 − 0.5i)23-s − 27-s + (−0.5 + 0.866i)29-s + (−1.36 + 0.366i)33-s + (0.866 − 0.5i)36-s + (1 + i)37-s + ⋯
L(s)  = 1  − 3-s + (0.866 − 0.5i)4-s + 9-s + (1.36 − 0.366i)11-s + (−0.866 + 0.5i)12-s + (0.5 + 0.866i)13-s + (0.499 − 0.866i)16-s − i·17-s + (−1 + i)19-s + (0.866 − 0.5i)23-s − 27-s + (−0.5 + 0.866i)29-s + (−1.36 + 0.366i)33-s + (0.866 − 0.5i)36-s + (1 + i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.919 + 0.393i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (2176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.919 + 0.393i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.250965346\)
\(L(\frac12)\) \(\approx\) \(1.250965346\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
13 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
7 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 + (1 - i)T - iT^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.866 + 0.5i)T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (-1 + i)T - iT^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
89 \( 1 + (-1 - i)T + iT^{2} \)
97 \( 1 + (-0.866 + 0.5i)T^{2} \)
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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

−9.124409765291113233380648230946, −8.031784457276360488624593519187, −6.90511310109354573344025656714, −6.64466356839560913280319171622, −6.04125062300452721235416149510, −5.14988178273420134382604218256, −4.30076231456971782866763306704, −3.33878965528293960305401460200, −1.91019180928855719358095150549, −1.12622466143218779620103461745, 1.21463547169001404971004794987, 2.24060520681680874517357841782, 3.58328680220920211981000917737, 4.16985925833762368292470743684, 5.25638626374358938827516945752, 6.27178089563315163664028425376, 6.48223551884187920856939801801, 7.33515845797171779467760492579, 8.086654665716087683620495688198, 8.987257285186787705309664075447

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