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Properties

Label 2-75-5.3-c2-0-5
Degree $2$
Conductor $75$
Sign $-0.229 + 0.973i$
Analytic cond. $2.04360$
Root an. cond. $1.42954$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.22 − 2.22i)2-s + (−1.22 − 1.22i)3-s − 5.89i·4-s − 5.44·6-s + (1.44 − 1.44i)7-s + (−4.22 − 4.22i)8-s + 2.99i·9-s − 3.34·11-s + (−7.22 + 7.22i)12-s + (10.4 + 10.4i)13-s − 6.44i·14-s + 4.79·16-s + (2.65 − 2.65i)17-s + (6.67 + 6.67i)18-s + 20.6i·19-s + ⋯
L(s)  = 1  + (1.11 − 1.11i)2-s + (−0.408 − 0.408i)3-s − 1.47i·4-s − 0.908·6-s + (0.207 − 0.207i)7-s + (−0.528 − 0.528i)8-s + 0.333i·9-s − 0.304·11-s + (−0.602 + 0.602i)12-s + (0.803 + 0.803i)13-s − 0.460i·14-s + 0.299·16-s + (0.155 − 0.155i)17-s + (0.370 + 0.370i)18-s + 1.08i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(2.04360\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.18278 - 1.49451i\)
\(L(\frac12)\) \(\approx\) \(1.18278 - 1.49451i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
good2 \( 1 + (-2.22 + 2.22i)T - 4iT^{2} \)
7 \( 1 + (-1.44 + 1.44i)T - 49iT^{2} \)
11 \( 1 + 3.34T + 121T^{2} \)
13 \( 1 + (-10.4 - 10.4i)T + 169iT^{2} \)
17 \( 1 + (-2.65 + 2.65i)T - 289iT^{2} \)
19 \( 1 - 20.6iT - 361T^{2} \)
23 \( 1 + (16.4 + 16.4i)T + 529iT^{2} \)
29 \( 1 + 0.853iT - 841T^{2} \)
31 \( 1 + 18.6T + 961T^{2} \)
37 \( 1 + (38.0 - 38.0i)T - 1.36e3iT^{2} \)
41 \( 1 + 28.6T + 1.68e3T^{2} \)
43 \( 1 + (22.4 + 22.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (19.7 - 19.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (28.6 + 28.6i)T + 2.80e3iT^{2} \)
59 \( 1 + 111. iT - 3.48e3T^{2} \)
61 \( 1 - 94.0T + 3.72e3T^{2} \)
67 \( 1 + (-54.8 + 54.8i)T - 4.48e3iT^{2} \)
71 \( 1 + 68T + 5.04e3T^{2} \)
73 \( 1 + (-39.7 - 39.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (-21.1 - 21.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 94.1iT - 7.92e3T^{2} \)
97 \( 1 + (14.5 - 14.5i)T - 9.40e3iT^{2} \)
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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

−13.78314428018357370070332041385, −12.76767858129223633791411356733, −11.90821052407197983036707136544, −11.04961397964289430059315854255, −10.05251592367988167095496601382, −8.162838991593512893350707408004, −6.41027452808946318855785491258, −5.07430020921464277112897775116, −3.69029309320002129007462159245, −1.77407759404490910354620685529, 3.61282662090980943200106223891, 5.06853230785096259032629481286, 5.92699959048462790542498919912, 7.25391599946350552797624846492, 8.566505160652055425661068781340, 10.26902422828552830293315898281, 11.54763086175669835155261392438, 12.81221175979910089960234644873, 13.63052971913073833368485735063, 14.80132117266502234524380837107

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