Properties

Label 2-75-25.6-c3-0-6
Degree $2$
Conductor $75$
Sign $0.852 - 0.521i$
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  + (0.536 + 1.65i)2-s + (−2.42 + 1.76i)3-s + (4.03 − 2.92i)4-s + (8.44 − 7.32i)5-s + (−4.21 − 3.06i)6-s + 7.66·7-s + (18.2 + 13.2i)8-s + (2.78 − 8.55i)9-s + (16.6 + 10.0i)10-s + (6.32 + 19.4i)11-s + (−4.61 + 14.2i)12-s + (−4.57 + 14.0i)13-s + (4.11 + 12.6i)14-s + (−7.58 + 32.6i)15-s + (0.205 − 0.633i)16-s + (88.1 + 64.0i)17-s + ⋯
L(s)  = 1  + (0.189 + 0.584i)2-s + (−0.467 + 0.339i)3-s + (0.503 − 0.366i)4-s + (0.755 − 0.655i)5-s + (−0.286 − 0.208i)6-s + 0.413·7-s + (0.806 + 0.585i)8-s + (0.103 − 0.317i)9-s + (0.526 + 0.316i)10-s + (0.173 + 0.533i)11-s + (−0.111 + 0.341i)12-s + (−0.0976 + 0.300i)13-s + (0.0785 + 0.241i)14-s + (−0.130 + 0.562i)15-s + (0.00321 − 0.00990i)16-s + (1.25 + 0.914i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.521i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.852 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.852 - 0.521i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ 0.852 - 0.521i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.81777 + 0.512031i\)
\(L(\frac12)\) \(\approx\) \(1.81777 + 0.512031i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.42 - 1.76i)T \)
5 \( 1 + (-8.44 + 7.32i)T \)
good2 \( 1 + (-0.536 - 1.65i)T + (-6.47 + 4.70i)T^{2} \)
7 \( 1 - 7.66T + 343T^{2} \)
11 \( 1 + (-6.32 - 19.4i)T + (-1.07e3 + 782. i)T^{2} \)
13 \( 1 + (4.57 - 14.0i)T + (-1.77e3 - 1.29e3i)T^{2} \)
17 \( 1 + (-88.1 - 64.0i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (79.1 + 57.4i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 + (24.8 + 76.6i)T + (-9.84e3 + 7.15e3i)T^{2} \)
29 \( 1 + (47.7 - 34.7i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (142. + 103. i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (71.2 - 219. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (-55.4 + 170. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 + 407.T + 7.95e4T^{2} \)
47 \( 1 + (386. - 280. i)T + (3.20e4 - 9.87e4i)T^{2} \)
53 \( 1 + (-417. + 303. i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (176. - 543. i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (16.1 + 49.7i)T + (-1.83e5 + 1.33e5i)T^{2} \)
67 \( 1 + (-446. - 324. i)T + (9.29e4 + 2.86e5i)T^{2} \)
71 \( 1 + (298. - 216. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (125. + 386. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (-529. + 384. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (341. + 247. i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 + (-131. - 404. i)T + (-5.70e5 + 4.14e5i)T^{2} \)
97 \( 1 + (477. - 347. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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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

−14.49505165692390889103513775565, −13.11598802913124314406397857521, −11.96114229213703796462987920173, −10.70456455864306587924650782557, −9.785573956784698726564708135245, −8.297978445413948796561346294163, −6.73236378038674553810103154048, −5.66192209143142240144432957754, −4.61511200884204902239227003059, −1.72496131987287211321492757778, 1.75089328746350227835567821503, 3.34875190113635617023782562799, 5.49024083431257161210617934380, 6.78380145590467467724182332604, 7.927307381734666454384289149036, 9.861123198649793758885761756466, 10.84899281478456285460355822122, 11.67584148415104392356963205942, 12.71484640528904382216997991461, 13.76513176119202246963116246397

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