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Properties

Label 2-75-25.11-c1-0-1
Degree $2$
Conductor $75$
Sign $0.728 - 0.684i$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (−0.309 + 0.951i)4-s + (0.690 + 2.12i)5-s + (0.309 + 0.951i)6-s + 4.47·7-s + (−0.927 − 2.85i)8-s + (−0.809 − 0.587i)9-s + (−1.80 − 1.31i)10-s + (−2.61 + 1.90i)11-s + (0.809 + 0.587i)12-s + (−2.73 − 1.98i)13-s + (−3.61 + 2.62i)14-s + 2.23·15-s + (0.809 + 0.587i)16-s + (−0.881 − 2.71i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.178 − 0.549i)3-s + (−0.154 + 0.475i)4-s + (0.309 + 0.951i)5-s + (0.126 + 0.388i)6-s + 1.69·7-s + (−0.327 − 1.00i)8-s + (−0.269 − 0.195i)9-s + (−0.572 − 0.415i)10-s + (−0.789 + 0.573i)11-s + (0.233 + 0.169i)12-s + (−0.758 − 0.551i)13-s + (−0.966 + 0.702i)14-s + 0.577·15-s + (0.202 + 0.146i)16-s + (−0.213 − 0.658i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.728 - 0.684i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1/2),\ 0.728 - 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738867 + 0.292538i\)
\(L(\frac12)\) \(\approx\) \(0.738867 + 0.292538i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.690 - 2.12i)T \)
good2 \( 1 + (0.809 - 0.587i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 + (2.61 - 1.90i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.73 + 1.98i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.881 + 2.71i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1 + 3.07i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.61 + 2.62i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.35 + 4.16i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.23 - 6.88i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.54 + 4.75i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.11 - 0.812i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.70T + 43T^{2} \)
47 \( 1 + (1.61 - 4.97i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.427 + 1.31i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.23 - 2.35i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.5 + 0.363i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.61 - 4.97i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (0.236 - 0.726i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.5 - 1.81i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.09 - 3.35i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (6.16 - 4.47i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.73 + 8.42i)T + (-78.4 - 57.0i)T^{2} \)
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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

−14.73993406725545392301282764623, −13.73578270710135781742699105557, −12.54363393248304636635882869051, −11.34542925508599531248887456893, −10.16111542054083921994065270493, −8.656260700052902557464099234712, −7.64740172922137002343071574524, −6.96459900789652105320208896956, −4.94157221680400257439315161765, −2.59762376004923137830598093600, 1.81840793571778125165229576876, 4.70036665276223795549786942922, 5.47576575030228824590906326445, 8.105639590580036310472066482838, 8.747487449512283935268750637188, 9.956435109683677422448013163879, 10.92882001834541987631384103838, 11.89044450322072216954771791340, 13.52510005910802409787809844874, 14.48269307418845588723424258529

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