Properties

Label 2-75-25.4-c1-0-1
Degree $2$
Conductor $75$
Sign $0.826 - 0.563i$
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  + (1.28 + 0.417i)2-s + (−0.587 + 0.809i)3-s + (−0.141 − 0.103i)4-s + (1.34 + 1.78i)5-s + (−1.09 + 0.793i)6-s − 1.59i·7-s + (−1.72 − 2.37i)8-s + (−0.309 − 0.951i)9-s + (0.980 + 2.85i)10-s + (1.02 − 3.16i)11-s + (0.166 − 0.0541i)12-s + (−6.70 + 2.17i)13-s + (0.666 − 2.05i)14-s + (−2.23 + 0.0363i)15-s + (−1.11 − 3.44i)16-s + (2.40 + 3.31i)17-s + ⋯
L(s)  = 1  + (0.908 + 0.295i)2-s + (−0.339 + 0.467i)3-s + (−0.0708 − 0.0515i)4-s + (0.600 + 0.799i)5-s + (−0.446 + 0.324i)6-s − 0.603i·7-s + (−0.610 − 0.840i)8-s + (−0.103 − 0.317i)9-s + (0.309 + 0.903i)10-s + (0.310 − 0.955i)11-s + (0.0481 − 0.0156i)12-s + (−1.85 + 0.604i)13-s + (0.178 − 0.547i)14-s + (−0.577 + 0.00939i)15-s + (−0.279 − 0.860i)16-s + (0.583 + 0.803i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.826 - 0.563i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1/2),\ 0.826 - 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17144 + 0.361108i\)
\(L(\frac12)\) \(\approx\) \(1.17144 + 0.361108i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (-1.34 - 1.78i)T \)
good2 \( 1 + (-1.28 - 0.417i)T + (1.61 + 1.17i)T^{2} \)
7 \( 1 + 1.59iT - 7T^{2} \)
11 \( 1 + (-1.02 + 3.16i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (6.70 - 2.17i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.40 - 3.31i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.459 + 0.333i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-5.99 - 1.94i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.25 + 1.63i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.805 + 0.585i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.37 - 1.09i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.359 - 1.10i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.117iT - 43T^{2} \)
47 \( 1 + (4.49 - 6.18i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.307 + 0.423i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.304 + 0.935i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.27 + 10.0i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (8.94 + 12.3i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-8.62 - 6.26i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-5.28 - 1.71i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (11.8 + 8.57i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.95 + 4.06i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.872 - 2.68i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-1.00 + 1.38i)T + (-29.9 - 92.2i)T^{2} \)
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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

−14.50282231300937342625589232984, −13.90331718348229920447660557296, −12.72190606756227018182994193976, −11.39388064758089027234985601060, −10.18788767464114972653602007130, −9.350474569479443145058725360390, −7.17303880380228773348081038057, −6.06318521808605348770510494852, −4.87986778235371478089080896749, −3.38793257218162412138455553644, 2.50384383699309512790639633675, 4.83638047766450803060667657237, 5.45114444048018811486099752740, 7.27212797247300556759577058363, 8.843603463914519364975971075176, 9.941487416710923344857068852642, 11.85039110802246264098408369319, 12.40691426585239356893515215400, 13.05338949302130369032338555976, 14.27898166861991272014184246791

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