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Properties

Label 2-75-3.2-c8-0-4
Degree $2$
Conductor $75$
Sign $0.863 - 0.503i$
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  − 18.2i·2-s + (−69.9 + 40.7i)3-s − 75.9·4-s + (743. + 1.27e3i)6-s − 4.67e3·7-s − 3.28e3i·8-s + (3.23e3 − 5.70e3i)9-s − 2.53e4i·11-s + (5.31e3 − 3.09e3i)12-s + 600.·13-s + 8.52e4i·14-s − 7.92e4·16-s + 7.32e4i·17-s + (−1.04e5 − 5.89e4i)18-s − 3.84e4·19-s + ⋯
L(s)  = 1  − 1.13i·2-s + (−0.863 + 0.503i)3-s − 0.296·4-s + (0.573 + 0.983i)6-s − 1.94·7-s − 0.801i·8-s + (0.492 − 0.870i)9-s − 1.72i·11-s + (0.256 − 0.149i)12-s + 0.0210·13-s + 2.21i·14-s − 1.20·16-s + 0.876i·17-s + (−0.990 − 0.561i)18-s − 0.294·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.863 - 0.503i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.863 - 0.503i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.375789 + 0.101523i\)
\(L(\frac12)\) \(\approx\) \(0.375789 + 0.101523i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (69.9 - 40.7i)T \)
5 \( 1 \)
good2 \( 1 + 18.2iT - 256T^{2} \)
7 \( 1 + 4.67e3T + 5.76e6T^{2} \)
11 \( 1 + 2.53e4iT - 2.14e8T^{2} \)
13 \( 1 - 600.T + 8.15e8T^{2} \)
17 \( 1 - 7.32e4iT - 6.97e9T^{2} \)
19 \( 1 + 3.84e4T + 1.69e10T^{2} \)
23 \( 1 - 1.96e5iT - 7.83e10T^{2} \)
29 \( 1 - 5.04e5iT - 5.00e11T^{2} \)
31 \( 1 - 6.65e5T + 8.52e11T^{2} \)
37 \( 1 + 3.51e5T + 3.51e12T^{2} \)
41 \( 1 - 2.88e6iT - 7.98e12T^{2} \)
43 \( 1 - 1.17e6T + 1.16e13T^{2} \)
47 \( 1 + 4.20e6iT - 2.38e13T^{2} \)
53 \( 1 + 6.57e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.46e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.09e7T + 1.91e14T^{2} \)
67 \( 1 + 1.82e7T + 4.06e14T^{2} \)
71 \( 1 - 1.95e7iT - 6.45e14T^{2} \)
73 \( 1 - 3.13e7T + 8.06e14T^{2} \)
79 \( 1 + 1.63e6T + 1.51e15T^{2} \)
83 \( 1 - 3.92e7iT - 2.25e15T^{2} \)
89 \( 1 + 1.89e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.16e8T + 7.83e15T^{2} \)
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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

−12.72558515501559106856094346110, −11.74460352910326133943761040437, −10.73757437202668706884800083196, −10.02789685205880695218831607573, −8.999712830678843718373403572187, −6.65091064192270406280827843230, −5.89392695490525969707235204820, −3.79454320040743435147952472864, −3.08651729851563801221615894075, −0.902568153977889926325433794099, 0.17313488402755529894851595706, 2.40154493689585325778227554911, 4.65869693924449879231078057032, 6.06722470477729006985574310442, 6.78075219288543193054622431318, 7.53033096696727644463550392124, 9.365961752211823387939394830480, 10.42420696675820446950201519481, 12.02773766420492095698194915098, 12.74569924799017406326406656418

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