Properties

Label 2-75-75.53-c1-0-1
Degree $2$
Conductor $75$
Sign $0.100 - 0.994i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.685 + 0.349i)2-s + (1.03 + 1.38i)3-s + (−0.827 + 1.13i)4-s + (−2.01 + 0.962i)5-s + (−1.19 − 0.587i)6-s + (1.53 − 1.53i)7-s + (0.410 − 2.58i)8-s + (−0.843 + 2.87i)9-s + (1.04 − 1.36i)10-s + (4.90 + 1.59i)11-s + (−2.43 + 0.0357i)12-s + (1.29 − 2.54i)13-s + (−0.517 + 1.59i)14-s + (−3.43 − 1.79i)15-s + (−0.246 − 0.759i)16-s + (−0.429 − 0.0680i)17-s + ⋯
L(s)  = 1  + (−0.484 + 0.246i)2-s + (0.599 + 0.800i)3-s + (−0.413 + 0.569i)4-s + (−0.902 + 0.430i)5-s + (−0.488 − 0.239i)6-s + (0.581 − 0.581i)7-s + (0.145 − 0.915i)8-s + (−0.281 + 0.959i)9-s + (0.331 − 0.431i)10-s + (1.47 + 0.480i)11-s + (−0.703 + 0.0103i)12-s + (0.360 − 0.706i)13-s + (−0.138 + 0.425i)14-s + (−0.885 − 0.464i)15-s + (−0.0617 − 0.189i)16-s + (−0.104 − 0.0165i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.100 - 0.994i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1/2),\ 0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.572541 + 0.517545i\)
\(L(\frac12)\) \(\approx\) \(0.572541 + 0.517545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.03 - 1.38i)T \)
5 \( 1 + (2.01 - 0.962i)T \)
good2 \( 1 + (0.685 - 0.349i)T + (1.17 - 1.61i)T^{2} \)
7 \( 1 + (-1.53 + 1.53i)T - 7iT^{2} \)
11 \( 1 + (-4.90 - 1.59i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.29 + 2.54i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.429 + 0.0680i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (0.215 + 0.297i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.51 + 4.93i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (0.866 + 0.629i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (7.68 - 5.58i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.68 - 1.36i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (-3.75 + 1.22i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (1.30 + 1.30i)T + 43iT^{2} \)
47 \( 1 + (1.03 + 6.51i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-1.97 + 0.312i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (1.69 + 5.21i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.56 + 10.9i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-0.319 + 2.01i)T + (-63.7 - 20.7i)T^{2} \)
71 \( 1 + (6.93 - 9.54i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (11.0 - 5.65i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-0.932 + 1.28i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.52 - 9.62i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (-2.35 + 7.24i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-2.72 + 0.431i)T + (92.2 - 29.9i)T^{2} \)
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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

−14.78884576357879577123145314265, −14.10700910503864251124246329482, −12.62479686426099848529242330272, −11.31099148033289744461080535409, −10.21046607417400607457144581581, −8.937565004226392577049454372040, −8.084187918742569129110479593008, −7.04343806738025275992851818522, −4.42753941035009730574710756613, −3.59345474141784736596862178897, 1.52137314880172834937092376119, 4.03690139876800973599972386641, 5.94977603306474409291102043670, 7.63340602089031222853346867452, 8.806297744718306929428207349362, 9.232477984224873182867906576993, 11.35544938165512993804881454021, 11.82669441075366225879737729277, 13.29080553603421616086902800272, 14.40823119662555679721638835029

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