Properties

Label 2-7500-5.4-c1-0-12
Degree $2$
Conductor $7500$
Sign $-i$
Analytic cond. $59.8878$
Root an. cond. $7.73872$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·3-s − 1.57i·7-s − 9-s + 3.88·11-s − 0.343i·13-s + 6.07i·17-s − 3.69·19-s − 1.57·21-s + 1.39i·23-s + i·27-s + 3.32·29-s − 5.25·31-s − 3.88i·33-s + 8.56i·37-s − 0.343·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.596i·7-s − 0.333·9-s + 1.17·11-s − 0.0952i·13-s + 1.47i·17-s − 0.846·19-s − 0.344·21-s + 0.289i·23-s + 0.192i·27-s + 0.618·29-s − 0.943·31-s − 0.675i·33-s + 1.40i·37-s − 0.0549·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7500\)    =    \(2^{2} \cdot 3 \cdot 5^{4}\)
Sign: $-i$
Analytic conductor: \(59.8878\)
Root analytic conductor: \(7.73872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7500} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7500,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9490912436\)
\(L(\frac12)\) \(\approx\) \(0.9490912436\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 + 1.57iT - 7T^{2} \)
11 \( 1 - 3.88T + 11T^{2} \)
13 \( 1 + 0.343iT - 13T^{2} \)
17 \( 1 - 6.07iT - 17T^{2} \)
19 \( 1 + 3.69T + 19T^{2} \)
23 \( 1 - 1.39iT - 23T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 + 5.25T + 31T^{2} \)
37 \( 1 - 8.56iT - 37T^{2} \)
41 \( 1 + 1.27T + 41T^{2} \)
43 \( 1 - 1.42iT - 43T^{2} \)
47 \( 1 - 0.375iT - 47T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 - 10.4iT - 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 13.1iT - 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + 4.62iT - 83T^{2} \)
89 \( 1 - 7.26T + 89T^{2} \)
97 \( 1 - 6.05iT - 97T^{2} \)
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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

−8.037168810434472199984596118059, −7.34711180189850790542903778719, −6.51406398132922047311916665388, −6.30583275002899544952393836445, −5.35295140565628591179470760982, −4.31014364749422526455125496384, −3.84850967591157169343202464495, −2.93106327623728960019079629690, −1.74839725355197382556541851432, −1.21666464501048701883911814256, 0.22511926489079059664572146396, 1.60357872993855803775456593766, 2.58545945421306852532877592760, 3.32993626978917596200677254719, 4.27896186925909090315350119377, 4.71679228771167038428529404664, 5.69331370643548957655463437109, 6.19305675497956137084120242700, 7.03785272082885420531773067656, 7.65161318240028123937892402387

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