Properties

Label 2-75-15.14-c8-0-18
Degree $2$
Conductor $75$
Sign $0.868 + 0.495i$
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  − 22.4·2-s + (−67.3 + 45i)3-s + 248·4-s + (1.51e3 − 1.01e3i)6-s + 1.75e3i·7-s + 179.·8-s + (2.51e3 − 6.06e3i)9-s − 6.95e3i·11-s + (−1.67e4 + 1.11e4i)12-s + 2.57e4i·13-s − 3.92e4i·14-s − 6.75e4·16-s − 7.48e4·17-s + (−5.63e4 + 1.36e5i)18-s − 1.89e4·19-s + ⋯
L(s)  = 1  − 1.40·2-s + (−0.831 + 0.555i)3-s + 0.968·4-s + (1.16 − 0.779i)6-s + 0.728i·7-s + 0.0438·8-s + (0.382 − 0.923i)9-s − 0.475i·11-s + (−0.805 + 0.538i)12-s + 0.900i·13-s − 1.02i·14-s − 1.03·16-s − 0.896·17-s + (−0.536 + 1.29i)18-s − 0.145·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.868 + 0.495i$
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: \((2,\ 75,\ (\ :4),\ 0.868 + 0.495i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.307347 - 0.0814514i\)
\(L(\frac12)\) \(\approx\) \(0.307347 - 0.0814514i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (67.3 - 45i)T \)
5 \( 1 \)
good2 \( 1 + 22.4T + 256T^{2} \)
7 \( 1 - 1.75e3iT - 5.76e6T^{2} \)
11 \( 1 + 6.95e3iT - 2.14e8T^{2} \)
13 \( 1 - 2.57e4iT - 8.15e8T^{2} \)
17 \( 1 + 7.48e4T + 6.97e9T^{2} \)
19 \( 1 + 1.89e4T + 1.69e10T^{2} \)
23 \( 1 + 4.70e5T + 7.83e10T^{2} \)
29 \( 1 - 4.60e5iT - 5.00e11T^{2} \)
31 \( 1 + 3.51e5T + 8.52e11T^{2} \)
37 \( 1 + 1.33e6iT - 3.51e12T^{2} \)
41 \( 1 + 1.87e6iT - 7.98e12T^{2} \)
43 \( 1 + 3.52e6iT - 1.16e13T^{2} \)
47 \( 1 - 4.08e6T + 2.38e13T^{2} \)
53 \( 1 + 6.60e6T + 6.22e13T^{2} \)
59 \( 1 - 1.37e7iT - 1.46e14T^{2} \)
61 \( 1 - 7.53e5T + 1.91e14T^{2} \)
67 \( 1 + 2.26e6iT - 4.06e14T^{2} \)
71 \( 1 + 1.70e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.76e7iT - 8.06e14T^{2} \)
79 \( 1 - 2.29e7T + 1.51e15T^{2} \)
83 \( 1 + 4.63e7T + 2.25e15T^{2} \)
89 \( 1 - 7.26e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.47e8iT - 7.83e15T^{2} \)
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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

−12.28684290326499868337782766031, −11.31555044233810425050916193094, −10.45082110408543680576143095898, −9.327035222889611915298808287029, −8.627534557001692721348142002575, −7.03147092777359259468538312687, −5.80066777165658366929060707671, −4.21012150461891235477603698052, −1.94596074272930092986494057165, −0.28752246922820041721980186388, 0.70112756587193685475036515377, 1.98675788241854501046629837510, 4.51398480959655594650912759312, 6.25971987182531883830799790650, 7.41730225025252755695091241705, 8.192975350988981454903177889740, 9.801980895748280923476553100146, 10.55331569360994764921540357719, 11.50030047079983131337415013922, 12.82026859361561444989090400491

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