Properties

Label 2-75-25.11-c3-0-5
Degree $2$
Conductor $75$
Sign $0.926 - 0.377i$
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  + (−3.16 + 2.29i)2-s + (0.927 − 2.85i)3-s + (2.24 − 6.90i)4-s + (−11.0 + 1.45i)5-s + (3.62 + 11.1i)6-s + 22.0·7-s + (−0.889 − 2.73i)8-s + (−7.28 − 5.29i)9-s + (31.6 − 30.0i)10-s + (31.5 − 22.9i)11-s + (−17.6 − 12.8i)12-s + (55.3 + 40.2i)13-s + (−69.8 + 50.7i)14-s + (−6.12 + 32.9i)15-s + (56.1 + 40.7i)16-s + (31.1 + 95.9i)17-s + ⋯
L(s)  = 1  + (−1.11 + 0.811i)2-s + (0.178 − 0.549i)3-s + (0.280 − 0.863i)4-s + (−0.991 + 0.130i)5-s + (0.246 + 0.758i)6-s + 1.19·7-s + (−0.0393 − 0.120i)8-s + (−0.269 − 0.195i)9-s + (1.00 − 0.950i)10-s + (0.866 − 0.629i)11-s + (−0.424 − 0.308i)12-s + (1.18 + 0.857i)13-s + (−1.33 + 0.968i)14-s + (−0.105 + 0.567i)15-s + (0.876 + 0.636i)16-s + (0.444 + 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.926 - 0.377i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ 0.926 - 0.377i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.863573 + 0.169104i\)
\(L(\frac12)\) \(\approx\) \(0.863573 + 0.169104i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.927 + 2.85i)T \)
5 \( 1 + (11.0 - 1.45i)T \)
good2 \( 1 + (3.16 - 2.29i)T + (2.47 - 7.60i)T^{2} \)
7 \( 1 - 22.0T + 343T^{2} \)
11 \( 1 + (-31.5 + 22.9i)T + (411. - 1.26e3i)T^{2} \)
13 \( 1 + (-55.3 - 40.2i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (-31.1 - 95.9i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (29.0 + 89.4i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 + (-130. + 94.8i)T + (3.75e3 - 1.15e4i)T^{2} \)
29 \( 1 + (15.8 - 48.7i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (80.4 + 247. i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (-70.1 - 50.9i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (-42.4 - 30.8i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 + 53.3T + 7.95e4T^{2} \)
47 \( 1 + (-74.7 + 230. i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (-18.2 + 56.2i)T + (-1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (-524. - 380. i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (530. - 385. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + (-240. - 740. i)T + (-2.43e5 + 1.76e5i)T^{2} \)
71 \( 1 + (-69.2 + 213. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (281. - 204. i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-49.9 + 153. i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (12.3 + 37.9i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 + (-375. + 272. i)T + (2.17e5 - 6.70e5i)T^{2} \)
97 \( 1 + (-177. + 546. i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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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

−14.59099120213715640979050357311, −13.07333317914998911805954433575, −11.60282059664785524816367053904, −10.87689521227982187678029883308, −8.775620051447755684223606953425, −8.496165688102319340403872511044, −7.31702730449992040336819234825, −6.28717209299482341687823067170, −3.98984210488494501789744638695, −1.10357717835259010762229796385, 1.24343165714397648885215034625, 3.45077885085894822439143589958, 5.07247614589122525819388726587, 7.59656117012036214405630350789, 8.465726041764517347970425740972, 9.420180010722948008283282480112, 10.78034619559340529107162281426, 11.36654261113322306457711881682, 12.31508996098057379422259420291, 14.22884176515390277480239078246

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