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Properties

Label 2-75-15.2-c7-0-26
Degree $2$
Conductor $75$
Sign $0.730 - 0.682i$
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 + (4.54 + 46.5i)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 + (6.48e3 − 633. i)12-s + (5.95e3 − 5.95e3i)13-s − 1.46e4·14-s + 1.47e4·16-s + (1.80e4 − 1.80e4i)17-s + (1.99e4 − 2.97e4i)18-s − 5.60e4i·19-s + ⋯
L(s)  = 1  + (−1.02 + 1.02i)2-s + (0.0971 + 0.995i)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 + (1.08 − 0.105i)12-s + (0.751 − 0.751i)13-s − 1.42·14-s + 0.903·16-s + (0.889 − 0.889i)17-s + (0.805 − 1.20i)18-s − 1.87i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.730 - 0.682i$
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: \((2,\ 75,\ (\ :7/2),\ 0.730 - 0.682i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.901946 + 0.355851i\)
\(L(\frac12)\) \(\approx\) \(0.901946 + 0.355851i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.54 - 46.5i)T \)
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

−13.62324618820313556608957628708, −11.74390384108959580092898192953, −10.70236693996717416996857653520, −9.439267212623857461958849722543, −8.675971151973411157639909981063, −7.79289690010718450549940436516, −6.08971971508371155132872907139, −5.04487641083915677150769218372, −3.06522236916551152473152069539, −0.52693738831085901707307128727, 1.23784409120956001678776619326, 1.83417068281189441529663115595, 3.69884909990440331555336042950, 5.98307722383599878494902638500, 7.66101598310070167626097691271, 8.305208774934152219572329471750, 9.704733581119548997106226852921, 10.77658150992034159845118414446, 11.76719114501935726042912568887, 12.54913378232990582214713388037

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