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Properties

Label 2-2475-2475.758-c0-0-0
Degree $2$
Conductor $2475$
Sign $0.315 - 0.948i$
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  + (0.104 − 0.994i)3-s + (−0.406 + 0.913i)4-s + (−0.866 − 0.5i)5-s + (−0.978 − 0.207i)9-s + (−0.207 + 0.978i)11-s + (0.866 + 0.499i)12-s + (−0.587 + 0.809i)15-s + (−0.669 − 0.743i)16-s + (0.809 − 0.587i)20-s + (0.0163 + 0.312i)23-s + (0.499 + 0.866i)25-s + (−0.309 + 0.951i)27-s + (−0.155 + 1.47i)31-s + (0.951 + 0.309i)33-s + (0.587 − 0.809i)36-s + (0.877 + 1.72i)37-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)3-s + (−0.406 + 0.913i)4-s + (−0.866 − 0.5i)5-s + (−0.978 − 0.207i)9-s + (−0.207 + 0.978i)11-s + (0.866 + 0.499i)12-s + (−0.587 + 0.809i)15-s + (−0.669 − 0.743i)16-s + (0.809 − 0.587i)20-s + (0.0163 + 0.312i)23-s + (0.499 + 0.866i)25-s + (−0.309 + 0.951i)27-s + (−0.155 + 1.47i)31-s + (0.951 + 0.309i)33-s + (0.587 − 0.809i)36-s + (0.877 + 1.72i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.315 - 0.948i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (758, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ 0.315 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5917250636\)
\(L(\frac12)\) \(\approx\) \(0.5917250636\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.104 + 0.994i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.207 - 0.978i)T \)
good2 \( 1 + (0.406 - 0.913i)T^{2} \)
7 \( 1 + (0.866 - 0.5i)T^{2} \)
13 \( 1 + (-0.406 - 0.913i)T^{2} \)
17 \( 1 + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.0163 - 0.312i)T + (-0.994 + 0.104i)T^{2} \)
29 \( 1 + (0.978 - 0.207i)T^{2} \)
31 \( 1 + (0.155 - 1.47i)T + (-0.978 - 0.207i)T^{2} \)
37 \( 1 + (-0.877 - 1.72i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (0.913 - 0.406i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.325 + 0.402i)T + (-0.207 - 0.978i)T^{2} \)
53 \( 1 + (0.0163 - 0.103i)T + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (1.91 - 0.406i)T + (0.913 - 0.406i)T^{2} \)
61 \( 1 + (-0.913 - 0.406i)T^{2} \)
67 \( 1 + (-0.978 - 1.20i)T + (-0.207 + 0.978i)T^{2} \)
71 \( 1 + (-0.244 - 0.336i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.587 + 0.809i)T^{2} \)
79 \( 1 + (-0.978 + 0.207i)T^{2} \)
83 \( 1 + (0.743 + 0.669i)T^{2} \)
89 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-1.55 - 1.25i)T + (0.207 + 0.978i)T^{2} \)
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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

−8.949857243682068566692270856039, −8.365717591357422421969428874604, −7.72203167107014077851281119800, −7.24371896209734512885712634918, −6.46423837792344644120274370117, −5.15412341289756159736239745607, −4.52238926340186681257414458309, −3.49979152429542466384924095338, −2.70722772833095860429521822339, −1.36015932614686688360919043925, 0.41298191324109658707306040801, 2.37920925529314497314189298321, 3.44237453259546844766539374693, 4.15351735776815112929115285481, 4.91274617842877011936989230551, 5.82377863632341894035180675849, 6.37309239837758087903524730013, 7.65631918603145984087013017768, 8.287033814702191722839574000554, 9.121962720733973682938781292448

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