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Properties

Label 2-75-15.2-c7-0-13
Degree $2$
Conductor $75$
Sign $-0.669 - 0.743i$
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  + (5.42 − 5.42i)2-s + (−13.1 + 44.8i)3-s + 69.0i·4-s + (172. + 314. i)6-s + (775. + 775. i)7-s + (1.06e3 + 1.06e3i)8-s + (−1.84e3 − 1.17e3i)9-s + 1.97e3i·11-s + (−3.10e3 − 905. i)12-s + (6.60e3 − 6.60e3i)13-s + 8.41e3·14-s + 2.77e3·16-s + (−2.33e4 + 2.33e4i)17-s + (−1.63e4 + 3.61e3i)18-s + 9.98e3i·19-s + ⋯
L(s)  = 1  + (0.479 − 0.479i)2-s + (−0.280 + 0.959i)3-s + 0.539i·4-s + (0.325 + 0.595i)6-s + (0.854 + 0.854i)7-s + (0.738 + 0.738i)8-s + (−0.842 − 0.538i)9-s + 0.448i·11-s + (−0.517 − 0.151i)12-s + (0.833 − 0.833i)13-s + 0.820·14-s + 0.169·16-s + (−1.15 + 1.15i)17-s + (−0.662 + 0.146i)18-s + 0.333i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.743i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.669 - 0.743i$
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: \((2,\ 75,\ (\ :7/2),\ -0.669 - 0.743i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.821691 + 1.84533i\)
\(L(\frac12)\) \(\approx\) \(0.821691 + 1.84533i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.1 - 44.8i)T \)
5 \( 1 \)
good2 \( 1 + (-5.42 + 5.42i)T - 128iT^{2} \)
7 \( 1 + (-775. - 775. i)T + 8.23e5iT^{2} \)
11 \( 1 - 1.97e3iT - 1.94e7T^{2} \)
13 \( 1 + (-6.60e3 + 6.60e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (2.33e4 - 2.33e4i)T - 4.10e8iT^{2} \)
19 \( 1 - 9.98e3iT - 8.93e8T^{2} \)
23 \( 1 + (2.45e4 + 2.45e4i)T + 3.40e9iT^{2} \)
29 \( 1 + 1.35e5T + 1.72e10T^{2} \)
31 \( 1 + 1.28e5T + 2.75e10T^{2} \)
37 \( 1 + (-2.74e5 - 2.74e5i)T + 9.49e10iT^{2} \)
41 \( 1 - 2.18e4iT - 1.94e11T^{2} \)
43 \( 1 + (8.22e4 - 8.22e4i)T - 2.71e11iT^{2} \)
47 \( 1 + (-4.00e5 + 4.00e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (-5.40e5 - 5.40e5i)T + 1.17e12iT^{2} \)
59 \( 1 + 1.84e6T + 2.48e12T^{2} \)
61 \( 1 - 1.57e6T + 3.14e12T^{2} \)
67 \( 1 + (7.81e5 + 7.81e5i)T + 6.06e12iT^{2} \)
71 \( 1 + 7.66e5iT - 9.09e12T^{2} \)
73 \( 1 + (-8.53e5 + 8.53e5i)T - 1.10e13iT^{2} \)
79 \( 1 + 6.22e6iT - 1.92e13T^{2} \)
83 \( 1 + (-4.42e6 - 4.42e6i)T + 2.71e13iT^{2} \)
89 \( 1 - 1.97e6T + 4.42e13T^{2} \)
97 \( 1 + (-7.15e6 - 7.15e6i)T + 8.07e13iT^{2} \)
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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

−13.30187740804899614376993530859, −12.26418318781051627229612546786, −11.29060468868738642654621152418, −10.56252732636473517071664491489, −8.903695171544785345590218055993, −8.060613723118616793644360038235, −5.92261501620384182766010738037, −4.71231160192618178806921008398, −3.63641703202702363070045349809, −2.09882008792101972639765133934, 0.60539630840262360253091229409, 1.82236642283097925562878966304, 4.26581848526350609478618042557, 5.55764109591414450432077917318, 6.73678202344804565566575304038, 7.58090455737347733784474464513, 9.110846617478141472127781244337, 10.94558009861276483045639459806, 11.38918959920498909850928082890, 13.17056899880354585628263369503

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