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Properties

Label 2-2527-133.111-c0-0-0
Degree $2$
Conductor $2527$
Sign $0.795 - 0.605i$
Analytic cond. $1.26113$
Root an. cond. $1.12300$
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  + (−0.900 − 0.755i)2-s + (0.0663 + 0.376i)4-s + (0.5 + 0.866i)7-s + (−0.363 + 0.629i)8-s + (0.766 − 0.642i)9-s + (−0.809 + 1.40i)11-s + (0.204 − 1.15i)14-s + (1.16 − 0.422i)16-s − 1.17·18-s + (1.78 − 0.650i)22-s + (0.107 + 0.608i)23-s + (−0.939 − 0.342i)25-s + (−0.292 + 0.245i)28-s + (−1.45 + 1.22i)29-s + (−0.682 − 0.248i)32-s + ⋯
L(s)  = 1  + (−0.900 − 0.755i)2-s + (0.0663 + 0.376i)4-s + (0.5 + 0.866i)7-s + (−0.363 + 0.629i)8-s + (0.766 − 0.642i)9-s + (−0.809 + 1.40i)11-s + (0.204 − 1.15i)14-s + (1.16 − 0.422i)16-s − 1.17·18-s + (1.78 − 0.650i)22-s + (0.107 + 0.608i)23-s + (−0.939 − 0.342i)25-s + (−0.292 + 0.245i)28-s + (−1.45 + 1.22i)29-s + (−0.682 − 0.248i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $0.795 - 0.605i$
Analytic conductor: \(1.26113\)
Root analytic conductor: \(1.12300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (776, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2527,\ (\ :0),\ 0.795 - 0.605i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6058728070\)
\(L(\frac12)\) \(\approx\) \(0.6058728070\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 \)
good2 \( 1 + (0.900 + 0.755i)T + (0.173 + 0.984i)T^{2} \)
3 \( 1 + (-0.766 + 0.642i)T^{2} \)
5 \( 1 + (0.939 + 0.342i)T^{2} \)
11 \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.107 - 0.608i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (1.45 - 1.22i)T + (0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.280 - 1.59i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.204 - 1.15i)T + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.900 + 0.755i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (-0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (1.78 - 0.650i)T + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.173 - 0.984i)T^{2} \)
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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

−9.495650053284407046234779521312, −8.616412682813509235313440657829, −7.77098878256178900356833272968, −7.20747330103403213710339581028, −5.99980073909196841006028727454, −5.24340734445977928883056140310, −4.41438657873157692232561628083, −3.16050318975612586133449556998, −2.10872161470269945116952174996, −1.50835399447251864808311032119, 0.54799054388455128592619107518, 1.99343883236253762022558527924, 3.45037961244127303179324929929, 4.18223855637529392682741398559, 5.32826235392721940123549759477, 6.04798187794401438752886066510, 7.11713341183931971442352702308, 7.52840630251605424054916559591, 8.212571021345366222236001560197, 8.680223103771334158033163385020

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