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Properties

Label 2-75-15.14-c8-0-38
Degree $2$
Conductor $75$
Sign $0.996 + 0.0876i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 29.4·2-s + (75.3 − 29.7i)3-s + 612.·4-s + (2.22e3 − 876. i)6-s + 3.16e3i·7-s + 1.05e4·8-s + (4.79e3 − 4.48e3i)9-s − 2.01e4i·11-s + (4.61e4 − 1.82e4i)12-s + 3.14e4i·13-s + 9.32e4i·14-s + 1.53e5·16-s − 2.61e4·17-s + (1.41e5 − 1.32e5i)18-s + 1.27e5·19-s + ⋯
L(s)  = 1  + 1.84·2-s + (0.930 − 0.367i)3-s + 2.39·4-s + (1.71 − 0.676i)6-s + 1.31i·7-s + 2.56·8-s + (0.730 − 0.682i)9-s − 1.37i·11-s + (2.22 − 0.878i)12-s + 1.10i·13-s + 2.42i·14-s + 2.33·16-s − 0.313·17-s + (1.34 − 1.25i)18-s + 0.980·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0876i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.996 + 0.0876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.996 + 0.0876i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.996 + 0.0876i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(8.50867 - 0.373583i\)
\(L(\frac12)\) \(\approx\) \(8.50867 - 0.373583i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-75.3 + 29.7i)T \)
5 \( 1 \)
good2 \( 1 - 29.4T + 256T^{2} \)
7 \( 1 - 3.16e3iT - 5.76e6T^{2} \)
11 \( 1 + 2.01e4iT - 2.14e8T^{2} \)
13 \( 1 - 3.14e4iT - 8.15e8T^{2} \)
17 \( 1 + 2.61e4T + 6.97e9T^{2} \)
19 \( 1 - 1.27e5T + 1.69e10T^{2} \)
23 \( 1 + 3.78e5T + 7.83e10T^{2} \)
29 \( 1 + 7.59e5iT - 5.00e11T^{2} \)
31 \( 1 + 8.32e5T + 8.52e11T^{2} \)
37 \( 1 + 1.09e6iT - 3.51e12T^{2} \)
41 \( 1 - 2.23e4iT - 7.98e12T^{2} \)
43 \( 1 - 4.00e6iT - 1.16e13T^{2} \)
47 \( 1 + 3.13e6T + 2.38e13T^{2} \)
53 \( 1 + 6.05e6T + 6.22e13T^{2} \)
59 \( 1 - 1.74e6iT - 1.46e14T^{2} \)
61 \( 1 - 7.86e6T + 1.91e14T^{2} \)
67 \( 1 + 9.88e6iT - 4.06e14T^{2} \)
71 \( 1 + 5.43e6iT - 6.45e14T^{2} \)
73 \( 1 - 5.05e7iT - 8.06e14T^{2} \)
79 \( 1 - 4.35e7T + 1.51e15T^{2} \)
83 \( 1 + 4.69e7T + 2.25e15T^{2} \)
89 \( 1 + 1.88e7iT - 3.93e15T^{2} \)
97 \( 1 + 3.52e7iT - 7.83e15T^{2} \)
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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

−13.10836879069080683312848154332, −12.04001713778768799300084528751, −11.37994235819648502010486074038, −9.343239031181775639484666043593, −8.026996798536447741891661954967, −6.51880964492051168818280595930, −5.61955602030198217410485684864, −4.02683274430209240996339344852, −2.90775262669923115237622411464, −1.90290167782207222140508485097, 1.79663009250410634822254655739, 3.24994515562967909391650963125, 4.15157001784744034032198398574, 5.16164774409475379772254922597, 6.98535729519667431257252834579, 7.73293412208124042327569651522, 9.924835463934590501364188799775, 10.76539862898969016474054558086, 12.30385514840908019070213058846, 13.18211749911223333620004771405

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