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Properties

Label 2-75-15.2-c9-0-16
Degree $2$
Conductor $75$
Sign $0.377 - 0.926i$
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  + (−29.6 + 29.6i)2-s + (−70.8 − 121. i)3-s − 1.24e3i·4-s + (5.68e3 + 1.48e3i)6-s + (7.25e3 + 7.25e3i)7-s + (2.17e4 + 2.17e4i)8-s + (−9.64e3 + 1.71e4i)9-s − 1.61e4i·11-s + (−1.50e5 + 8.82e4i)12-s + (1.18e5 − 1.18e5i)13-s − 4.30e5·14-s − 6.50e5·16-s + (−1.99e5 + 1.99e5i)17-s + (−2.22e5 − 7.94e5i)18-s − 4.94e4i·19-s + ⋯
L(s)  = 1  + (−1.30 + 1.30i)2-s + (−0.505 − 0.863i)3-s − 2.43i·4-s + (1.79 + 0.468i)6-s + (1.14 + 1.14i)7-s + (1.87 + 1.87i)8-s + (−0.489 + 0.871i)9-s − 0.332i·11-s + (−2.09 + 1.22i)12-s + (1.15 − 1.15i)13-s − 2.99·14-s − 2.48·16-s + (−0.579 + 0.579i)17-s + (−0.500 − 1.78i)18-s − 0.0870i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.377 - 0.926i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ 0.377 - 0.926i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.724390 + 0.486977i\)
\(L(\frac12)\) \(\approx\) \(0.724390 + 0.486977i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (70.8 + 121. i)T \)
5 \( 1 \)
good2 \( 1 + (29.6 - 29.6i)T - 512iT^{2} \)
7 \( 1 + (-7.25e3 - 7.25e3i)T + 4.03e7iT^{2} \)
11 \( 1 + 1.61e4iT - 2.35e9T^{2} \)
13 \( 1 + (-1.18e5 + 1.18e5i)T - 1.06e10iT^{2} \)
17 \( 1 + (1.99e5 - 1.99e5i)T - 1.18e11iT^{2} \)
19 \( 1 + 4.94e4iT - 3.22e11T^{2} \)
23 \( 1 + (-4.35e5 - 4.35e5i)T + 1.80e12iT^{2} \)
29 \( 1 - 9.90e4T + 1.45e13T^{2} \)
31 \( 1 + 7.60e6T + 2.64e13T^{2} \)
37 \( 1 + (-7.35e5 - 7.35e5i)T + 1.29e14iT^{2} \)
41 \( 1 + 2.84e7iT - 3.27e14T^{2} \)
43 \( 1 + (-4.64e6 + 4.64e6i)T - 5.02e14iT^{2} \)
47 \( 1 + (2.19e7 - 2.19e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (-4.34e7 - 4.34e7i)T + 3.29e15iT^{2} \)
59 \( 1 - 1.37e8T + 8.66e15T^{2} \)
61 \( 1 - 7.42e7T + 1.16e16T^{2} \)
67 \( 1 + (-4.34e7 - 4.34e7i)T + 2.72e16iT^{2} \)
71 \( 1 - 2.05e8iT - 4.58e16T^{2} \)
73 \( 1 + (1.81e8 - 1.81e8i)T - 5.88e16iT^{2} \)
79 \( 1 + 2.98e7iT - 1.19e17T^{2} \)
83 \( 1 + (2.00e8 + 2.00e8i)T + 1.86e17iT^{2} \)
89 \( 1 - 3.63e8T + 3.50e17T^{2} \)
97 \( 1 + (-6.54e8 - 6.54e8i)T + 7.60e17iT^{2} \)
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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

−12.97448195622804326867401313491, −11.37960858138842220952772799623, −10.63927225147087200585776616106, −8.772888361392199420977592862181, −8.335704387856637740033651322944, −7.25861890629992535464955219814, −5.88923480017935377347394009260, −5.41589848283665586715665650944, −1.92515196730502778492921462966, −0.813609854379454156529613178611, 0.64015873606788055848427150114, 1.77200508311885050979838038608, 3.65927686483942189967139184958, 4.57280311669518843878086319649, 6.99475920733492443308963317016, 8.394072872423154295938135855919, 9.335889140061502943246255906247, 10.40369907052242134343311334667, 11.23278900229219675914508221503, 11.58585238803240654336663769991

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