Properties

Label 2-75-15.8-c9-0-24
Degree $2$
Conductor $75$
Sign $-0.979 + 0.200i$
Analytic cond. $38.6276$
Root an. cond. $6.21511$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (27.8 + 27.8i)2-s + (131. + 48.2i)3-s + 1.03e3i·4-s + (2.32e3 + 5.00e3i)6-s + (−6.30e3 + 6.30e3i)7-s + (−1.45e4 + 1.45e4i)8-s + (1.50e4 + 1.27e4i)9-s − 1.00e4i·11-s + (−5.00e4 + 1.36e5i)12-s + (1.13e5 + 1.13e5i)13-s − 3.50e5·14-s − 2.80e5·16-s + (−4.18e5 − 4.18e5i)17-s + (6.41e4 + 7.71e5i)18-s − 7.43e4i·19-s + ⋯
L(s)  = 1  + (1.22 + 1.22i)2-s + (0.938 + 0.344i)3-s + 2.02i·4-s + (0.731 + 1.57i)6-s + (−0.991 + 0.991i)7-s + (−1.25 + 1.25i)8-s + (0.763 + 0.646i)9-s − 0.206i·11-s + (−0.696 + 1.89i)12-s + (1.10 + 1.10i)13-s − 2.43·14-s − 1.07·16-s + (−1.21 − 1.21i)17-s + (0.144 + 1.73i)18-s − 0.130i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.979 + 0.200i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ -0.979 + 0.200i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.482529 - 4.75300i\)
\(L(\frac12)\) \(\approx\) \(0.482529 - 4.75300i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-131. - 48.2i)T \)
5 \( 1 \)
good2 \( 1 + (-27.8 - 27.8i)T + 512iT^{2} \)
7 \( 1 + (6.30e3 - 6.30e3i)T - 4.03e7iT^{2} \)
11 \( 1 + 1.00e4iT - 2.35e9T^{2} \)
13 \( 1 + (-1.13e5 - 1.13e5i)T + 1.06e10iT^{2} \)
17 \( 1 + (4.18e5 + 4.18e5i)T + 1.18e11iT^{2} \)
19 \( 1 + 7.43e4iT - 3.22e11T^{2} \)
23 \( 1 + (-1.15e6 + 1.15e6i)T - 1.80e12iT^{2} \)
29 \( 1 + 5.61e6T + 1.45e13T^{2} \)
31 \( 1 + 1.72e6T + 2.64e13T^{2} \)
37 \( 1 + (5.75e6 - 5.75e6i)T - 1.29e14iT^{2} \)
41 \( 1 - 1.54e7iT - 3.27e14T^{2} \)
43 \( 1 + (-2.43e7 - 2.43e7i)T + 5.02e14iT^{2} \)
47 \( 1 + (-8.82e6 - 8.82e6i)T + 1.11e15iT^{2} \)
53 \( 1 + (-4.01e7 + 4.01e7i)T - 3.29e15iT^{2} \)
59 \( 1 - 1.04e8T + 8.66e15T^{2} \)
61 \( 1 - 1.51e7T + 1.16e16T^{2} \)
67 \( 1 + (-5.07e7 + 5.07e7i)T - 2.72e16iT^{2} \)
71 \( 1 + 4.26e7iT - 4.58e16T^{2} \)
73 \( 1 + (3.98e7 + 3.98e7i)T + 5.88e16iT^{2} \)
79 \( 1 - 1.98e8iT - 1.19e17T^{2} \)
83 \( 1 + (1.82e7 - 1.82e7i)T - 1.86e17iT^{2} \)
89 \( 1 - 7.56e8T + 3.50e17T^{2} \)
97 \( 1 + (-8.05e8 + 8.05e8i)T - 7.60e17iT^{2} \)
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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

−13.40181601995665707971489757499, −12.87722070325215309820316482663, −11.36779166943740860595732197346, −9.292662634251043053499222831199, −8.659952376219734270524103250988, −7.11294782317142815492729240718, −6.22695763760732550286358078803, −4.81807836381641872279588835630, −3.65577712457708649882143707406, −2.54265314462445184281428026226, 0.796181334392890507614566217595, 2.06308324040034381686274468514, 3.51582456751910778018386734991, 3.89074543104492749411943234378, 5.83845732324428750553162390023, 7.22048333050381154778655699228, 8.941045928587696476888905865541, 10.25178422776457098815155509130, 10.96330499996625843498741279804, 12.62104198686948306705209728545

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