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Properties

Label 2-75-15.14-c8-0-24
Degree $2$
Conductor $75$
Sign $-0.325 + 0.945i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.2·2-s + (−56.7 − 57.8i)3-s − 151.·4-s + (580. + 591. i)6-s − 3.44e3i·7-s + 4.16e3·8-s + (−122. + 6.55e3i)9-s − 7.12e3i·11-s + (8.58e3 + 8.74e3i)12-s + 4.13e4i·13-s + 3.52e4i·14-s − 3.95e3·16-s + 1.19e5·17-s + (1.25e3 − 6.71e4i)18-s + 8.62e4·19-s + ⋯
L(s)  = 1  − 0.639·2-s + (−0.700 − 0.713i)3-s − 0.590·4-s + (0.448 + 0.456i)6-s − 1.43i·7-s + 1.01·8-s + (−0.0187 + 0.999i)9-s − 0.486i·11-s + (0.413 + 0.421i)12-s + 1.44i·13-s + 0.918i·14-s − 0.0602·16-s + 1.43·17-s + (0.0119 − 0.639i)18-s + 0.662·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.325 + 0.945i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ -0.325 + 0.945i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.480180 - 0.672838i\)
\(L(\frac12)\) \(\approx\) \(0.480180 - 0.672838i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (56.7 + 57.8i)T \)
5 \( 1 \)
good2 \( 1 + 10.2T + 256T^{2} \)
7 \( 1 + 3.44e3iT - 5.76e6T^{2} \)
11 \( 1 + 7.12e3iT - 2.14e8T^{2} \)
13 \( 1 - 4.13e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.19e5T + 6.97e9T^{2} \)
19 \( 1 - 8.62e4T + 1.69e10T^{2} \)
23 \( 1 - 3.17e5T + 7.83e10T^{2} \)
29 \( 1 + 5.98e4iT - 5.00e11T^{2} \)
31 \( 1 + 1.02e6T + 8.52e11T^{2} \)
37 \( 1 + 8.77e5iT - 3.51e12T^{2} \)
41 \( 1 + 1.55e6iT - 7.98e12T^{2} \)
43 \( 1 + 2.56e6iT - 1.16e13T^{2} \)
47 \( 1 - 8.98e6T + 2.38e13T^{2} \)
53 \( 1 + 6.22e6T + 6.22e13T^{2} \)
59 \( 1 + 3.96e6iT - 1.46e14T^{2} \)
61 \( 1 + 5.63e5T + 1.91e14T^{2} \)
67 \( 1 + 1.34e7iT - 4.06e14T^{2} \)
71 \( 1 - 3.56e7iT - 6.45e14T^{2} \)
73 \( 1 - 9.70e5iT - 8.06e14T^{2} \)
79 \( 1 - 2.78e7T + 1.51e15T^{2} \)
83 \( 1 - 2.25e7T + 2.25e15T^{2} \)
89 \( 1 - 5.73e7iT - 3.93e15T^{2} \)
97 \( 1 - 3.31e7iT - 7.83e15T^{2} \)
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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

−12.56800378900069050737180581878, −11.23522853403879232903736140853, −10.38002954595419624992938942862, −9.160009529356933705364349888417, −7.69578225928250350504287322986, −7.01198443143739380072665131800, −5.30969523967388711024119985212, −3.93066479550509738478584809528, −1.38489228808479627488199734731, −0.53361105122599892292477641338, 0.963744591558344784446948117642, 3.21560083470266677574504873330, 5.01935988696801219285690970745, 5.69327390948119651166423258632, 7.69307416790668317115279698883, 8.973027826697457354404430451107, 9.746932191421959866390902926711, 10.72412017384345285379188723266, 12.08002283812356431821995861264, 12.87804504993388379438170782125

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