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Properties

Label 2-2475-165.98-c0-0-0
Degree $2$
Conductor $2475$
Sign $-0.998 - 0.0618i$
Analytic cond. $1.23518$
Root an. cond. $1.11138$
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  + (−1.30 + 1.30i)2-s − 2.41i·4-s + (1.30 + 1.30i)7-s + (1.84 + 1.84i)8-s + i·11-s + (−0.541 + 0.541i)13-s − 3.41·14-s − 2.41·16-s + (−0.541 + 0.541i)17-s + (−1.30 − 1.30i)22-s − 1.41i·26-s + (3.15 − 3.15i)28-s − 1.41·31-s + (1.30 − 1.30i)32-s − 1.41i·34-s + ⋯
L(s)  = 1  + (−1.30 + 1.30i)2-s − 2.41i·4-s + (1.30 + 1.30i)7-s + (1.84 + 1.84i)8-s + i·11-s + (−0.541 + 0.541i)13-s − 3.41·14-s − 2.41·16-s + (−0.541 + 0.541i)17-s + (−1.30 − 1.30i)22-s − 1.41i·26-s + (3.15 − 3.15i)28-s − 1.41·31-s + (1.30 − 1.30i)32-s − 1.41i·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.998 - 0.0618i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ -0.998 - 0.0618i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5822914078\)
\(L(\frac12)\) \(\approx\) \(0.5822914078\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 - iT \)
good2 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
7 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
13 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
17 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.969415239948821280139248417414, −8.895187136388233813810158858958, −7.922571076410726337939132066561, −7.40228129106809915705114558915, −6.63651416397258945035687474534, −5.71544130435856815309047991724, −5.15208369382608252189800019867, −4.35972686522240371958007260964, −2.22718725748870408080762175142, −1.67437467459614384415566365133, 0.61806714762758565946178062062, 1.59365133576891881507942923276, 2.64821200985332310567864560363, 3.63903271889435471938037318270, 4.39027265073625677684542111894, 5.43011880654021234982095405399, 6.98011448857517248932539613256, 7.55667893662220146706900403869, 8.169575581766859839732288355760, 8.784332020585509498225187529768

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