Properties

Label 2-75-15.2-c3-0-11
Degree $2$
Conductor $75$
Sign $0.432 + 0.901i$
Analytic cond. $4.42514$
Root an. cond. $2.10360$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.66 + 2.66i)2-s + (2.80 − 4.37i)3-s − 6.17i·4-s + (4.17 + 19.1i)6-s + (−9.35 − 9.35i)7-s + (−4.84 − 4.84i)8-s + (−11.2 − 24.5i)9-s − 34.1i·11-s + (−27.0 − 17.3i)12-s + (−2.82 + 2.82i)13-s + 49.8·14-s + 75.2·16-s + (64.2 − 64.2i)17-s + (95.3 + 35.3i)18-s − 19.0i·19-s + ⋯
L(s)  = 1  + (−0.941 + 0.941i)2-s + (0.539 − 0.841i)3-s − 0.772i·4-s + (0.284 + 1.30i)6-s + (−0.505 − 0.505i)7-s + (−0.214 − 0.214i)8-s + (−0.417 − 0.908i)9-s − 0.935i·11-s + (−0.650 − 0.416i)12-s + (−0.0601 + 0.0601i)13-s + 0.951·14-s + 1.17·16-s + (0.916 − 0.916i)17-s + (1.24 + 0.462i)18-s − 0.230i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.432 + 0.901i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ 0.432 + 0.901i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.672770 - 0.423545i\)
\(L(\frac12)\) \(\approx\) \(0.672770 - 0.423545i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.80 + 4.37i)T \)
5 \( 1 \)
good2 \( 1 + (2.66 - 2.66i)T - 8iT^{2} \)
7 \( 1 + (9.35 + 9.35i)T + 343iT^{2} \)
11 \( 1 + 34.1iT - 1.33e3T^{2} \)
13 \( 1 + (2.82 - 2.82i)T - 2.19e3iT^{2} \)
17 \( 1 + (-64.2 + 64.2i)T - 4.91e3iT^{2} \)
19 \( 1 + 19.0iT - 6.85e3T^{2} \)
23 \( 1 + (51.4 + 51.4i)T + 1.21e4iT^{2} \)
29 \( 1 + 50.5T + 2.43e4T^{2} \)
31 \( 1 + 93.3T + 2.97e4T^{2} \)
37 \( 1 + (-161. - 161. i)T + 5.06e4iT^{2} \)
41 \( 1 - 88.7iT - 6.89e4T^{2} \)
43 \( 1 + (176. - 176. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-38.2 + 38.2i)T - 1.03e5iT^{2} \)
53 \( 1 + (-344. - 344. i)T + 1.48e5iT^{2} \)
59 \( 1 - 421.T + 2.05e5T^{2} \)
61 \( 1 - 2T + 2.26e5T^{2} \)
67 \( 1 + (430. + 430. i)T + 3.00e5iT^{2} \)
71 \( 1 - 733. iT - 3.57e5T^{2} \)
73 \( 1 + (-348. + 348. i)T - 3.89e5iT^{2} \)
79 \( 1 + 588. iT - 4.93e5T^{2} \)
83 \( 1 + (217. + 217. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.27e3T + 7.04e5T^{2} \)
97 \( 1 + (-432. - 432. i)T + 9.12e5iT^{2} \)
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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

−13.97123696714236720853695931813, −12.98107638686373524948273957915, −11.72810623435533970211496748737, −9.993332843393581404867411373026, −8.958686783004563772680294890179, −7.942573121815940493406255962064, −7.04714702972692765239526535906, −5.99240906169719464558846470812, −3.26210922607426338884168119725, −0.64713994843582589697392965528, 2.13599406242055927338492735157, 3.64026316772527594457533895609, 5.58805430216827223400052700385, 7.82056389333882202765881623332, 8.982340030983961605989256172501, 9.815923577087376180915875927329, 10.49976368988746579773542752262, 11.79300209595299664603721158761, 12.83726292504414001786176792958, 14.48524279316479649925639510586

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