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Properties

Label 2-75-15.2-c7-0-23
Degree $2$
Conductor $75$
Sign $0.528 + 0.848i$
Analytic cond. $23.4288$
Root an. cond. $4.84033$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.5 − 11.5i)2-s + (−46.5 − 4.54i)3-s − 139. i·4-s + (−590. + 485. i)6-s + (632. + 632. i)7-s + (−131. − 131. i)8-s + (2.14e3 + 423. i)9-s + 2.86e3i·11-s + (−633. + 6.48e3i)12-s + (5.95e3 − 5.95e3i)13-s + 1.46e4·14-s + 1.47e4·16-s + (−1.80e4 + 1.80e4i)17-s + (2.97e4 − 1.99e4i)18-s − 5.60e4i·19-s + ⋯
L(s)  = 1  + (1.02 − 1.02i)2-s + (−0.995 − 0.0971i)3-s − 1.08i·4-s + (−1.11 + 0.917i)6-s + (0.697 + 0.697i)7-s + (−0.0908 − 0.0908i)8-s + (0.981 + 0.193i)9-s + 0.649i·11-s + (−0.105 + 1.08i)12-s + (0.751 − 0.751i)13-s + 1.42·14-s + 0.903·16-s + (−0.889 + 0.889i)17-s + (1.20 − 0.805i)18-s − 1.87i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.528 + 0.848i$
Analytic conductor: \(23.4288\)
Root analytic conductor: \(4.84033\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :7/2),\ 0.528 + 0.848i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.47542 - 1.37469i\)
\(L(\frac12)\) \(\approx\) \(2.47542 - 1.37469i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (46.5 + 4.54i)T \)
5 \( 1 \)
good2 \( 1 + (-11.5 + 11.5i)T - 128iT^{2} \)
7 \( 1 + (-632. - 632. i)T + 8.23e5iT^{2} \)
11 \( 1 - 2.86e3iT - 1.94e7T^{2} \)
13 \( 1 + (-5.95e3 + 5.95e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (1.80e4 - 1.80e4i)T - 4.10e8iT^{2} \)
19 \( 1 + 5.60e4iT - 8.93e8T^{2} \)
23 \( 1 + (-2.05e4 - 2.05e4i)T + 3.40e9iT^{2} \)
29 \( 1 - 1.93e5T + 1.72e10T^{2} \)
31 \( 1 - 2.48e5T + 2.75e10T^{2} \)
37 \( 1 + (-8.97e4 - 8.97e4i)T + 9.49e10iT^{2} \)
41 \( 1 - 1.93e5iT - 1.94e11T^{2} \)
43 \( 1 + (2.49e5 - 2.49e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (4.38e4 - 4.38e4i)T - 5.06e11iT^{2} \)
53 \( 1 + (2.59e5 + 2.59e5i)T + 1.17e12iT^{2} \)
59 \( 1 - 2.92e6T + 2.48e12T^{2} \)
61 \( 1 + 9.06e5T + 3.14e12T^{2} \)
67 \( 1 + (-1.64e6 - 1.64e6i)T + 6.06e12iT^{2} \)
71 \( 1 - 3.70e6iT - 9.09e12T^{2} \)
73 \( 1 + (-2.11e5 + 2.11e5i)T - 1.10e13iT^{2} \)
79 \( 1 + 5.81e6iT - 1.92e13T^{2} \)
83 \( 1 + (-6.37e4 - 6.37e4i)T + 2.71e13iT^{2} \)
89 \( 1 + 1.03e7T + 4.42e13T^{2} \)
97 \( 1 + (-3.23e6 - 3.23e6i)T + 8.07e13iT^{2} \)
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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

−12.83925429775746362382907498713, −11.73901447391307366162835592122, −11.17680473996448673155786249316, −10.17726549262466707348304879443, −8.350290529358086053741320469364, −6.53670952327174654372694779495, −5.18890858550539079250469822883, −4.41807963406885781295597877942, −2.55542657483306406671412626926, −1.15206139756058913411284755959, 1.06381501485658580331184696457, 3.96874952046272984119770070325, 4.84413894956266139219133673511, 6.08680670595439951137794479742, 6.90470363839177498685003243399, 8.231213009820163223151702094511, 10.19566026681999515814286713134, 11.26583710314285650729918314685, 12.31533254102305655246656409925, 13.66153827793629578668052899868

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