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Properties

Label 2-975-65.64-c1-0-12
Degree $2$
Conductor $975$
Sign $0.735 - 0.677i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  − 1.73·2-s − i·3-s + 0.999·4-s + 1.73i·6-s + 3.46·7-s + 1.73·8-s − 9-s + 3.46i·11-s − 0.999i·12-s + (3.46 − i)13-s − 5.99·14-s − 5·16-s + 6i·17-s + 1.73·18-s + 3.46i·19-s + ⋯
L(s)  = 1  − 1.22·2-s − 0.577i·3-s + 0.499·4-s + 0.707i·6-s + 1.30·7-s + 0.612·8-s − 0.333·9-s + 1.04i·11-s − 0.288i·12-s + (0.960 − 0.277i)13-s − 1.60·14-s − 1.25·16-s + 1.45i·17-s + 0.408·18-s + 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.735 - 0.677i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.735 - 0.677i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.787789 + 0.307678i\)
\(L(\frac12)\) \(\approx\) \(0.787789 + 0.307678i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
13 \( 1 + (-3.46 + i)T \)
good2 \( 1 + 1.73T + 2T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + 6.92T + 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−10.17382915560659511609436619911, −9.043736554024452258027349804939, −8.349914302981787908353116207050, −7.87271055295619749368382899311, −7.11845692200993850486456884647, −6.00572074868094050742490502864, −4.89954428136918525572047208581, −3.83062572989672980532946405485, −1.87308812148669726974207992930, −1.41886578122504838935572110845, 0.65727379596241508119569242589, 2.01949109300450741961756600515, 3.55893939643553442788316678832, 4.71439483849654755009880177184, 5.43926176931379553489922524871, 6.78740258230412198515477967739, 7.75835673234362661887858281313, 8.451063041699805188405215656848, 9.027340407906823776951348993479, 9.692433137818640632024025240519

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