Properties

Label 2-75-25.21-c3-0-5
Degree $2$
Conductor $75$
Sign $0.993 + 0.110i$
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  + (−1.53 + 4.70i)2-s + (−2.42 − 1.76i)3-s + (−13.3 − 9.71i)4-s + (−9.83 − 5.31i)5-s + (12.0 − 8.73i)6-s + 28.2·7-s + (34.1 − 24.8i)8-s + (2.78 + 8.55i)9-s + (40.0 − 38.1i)10-s + (−3.06 + 9.43i)11-s + (15.3 + 47.1i)12-s + (−28.5 − 87.7i)13-s + (−43.2 + 133. i)14-s + (14.5 + 30.2i)15-s + (23.7 + 73.0i)16-s + (15.8 − 11.5i)17-s + ⋯
L(s)  = 1  + (−0.541 + 1.66i)2-s + (−0.467 − 0.339i)3-s + (−1.67 − 1.21i)4-s + (−0.879 − 0.475i)5-s + (0.817 − 0.594i)6-s + 1.52·7-s + (1.50 − 1.09i)8-s + (0.103 + 0.317i)9-s + (1.26 − 1.20i)10-s + (−0.0840 + 0.258i)11-s + (0.368 + 1.13i)12-s + (−0.608 − 1.87i)13-s + (−0.826 + 2.54i)14-s + (0.249 + 0.520i)15-s + (0.370 + 1.14i)16-s + (0.226 − 0.164i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.993 + 0.110i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ 0.993 + 0.110i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.649635 - 0.0360808i\)
\(L(\frac12)\) \(\approx\) \(0.649635 - 0.0360808i\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.83 + 5.31i)T \)
good2 \( 1 + (1.53 - 4.70i)T + (-6.47 - 4.70i)T^{2} \)
7 \( 1 - 28.2T + 343T^{2} \)
11 \( 1 + (3.06 - 9.43i)T + (-1.07e3 - 782. i)T^{2} \)
13 \( 1 + (28.5 + 87.7i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (-15.8 + 11.5i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (-62.2 + 45.2i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 + (-31.3 + 96.3i)T + (-9.84e3 - 7.15e3i)T^{2} \)
29 \( 1 + (175. + 127. i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (84.9 - 61.7i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (102. + 315. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-29.3 - 90.4i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 + 67.8T + 7.95e4T^{2} \)
47 \( 1 + (95.0 + 69.0i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (-180. - 130. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (-53.9 - 166. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (145. - 449. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + (-101. + 73.8i)T + (9.29e4 - 2.86e5i)T^{2} \)
71 \( 1 + (750. + 545. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (-31.2 + 96.2i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (238. + 173. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (-899. + 653. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 + (-39.4 + 121. i)T + (-5.70e5 - 4.14e5i)T^{2} \)
97 \( 1 + (-1.23e3 - 894. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−14.62326031804206645314411093870, −13.12867569575212541049492556969, −11.86227431990177133132039787670, −10.57045563231360101264034611874, −8.884336551390992966731409912467, −7.72218477241647515417234010398, −7.50070347210644432711701884915, −5.51774255704679017620454968489, −4.81649681050252270985237525965, −0.57067805206936034152447885873, 1.66043630414913995821204244049, 3.66411547826428296003908377847, 4.82105203579581108079956488296, 7.42349992482875930620873034446, 8.688119313518725087057787201522, 9.882257819581406603996089682916, 11.25151995308069488633364398260, 11.38429786409651140337987955416, 12.24347138415832075524124281164, 13.94901996781140994208033273266

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