Properties

Label 2-75-75.17-c1-0-3
Degree $2$
Conductor $75$
Sign $0.909 - 0.415i$
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  + (0.710 + 0.362i)2-s + (1.53 + 0.807i)3-s + (−0.801 − 1.10i)4-s + (−0.787 + 2.09i)5-s + (0.796 + 1.12i)6-s + (−2.94 − 2.94i)7-s + (−0.419 − 2.65i)8-s + (1.69 + 2.47i)9-s + (−1.31 + 1.20i)10-s + (0.782 − 0.254i)11-s + (−0.337 − 2.33i)12-s + (0.935 + 1.83i)13-s + (−1.02 − 3.15i)14-s + (−2.89 + 2.57i)15-s + (−0.181 + 0.557i)16-s + (−2.17 + 0.344i)17-s + ⋯
L(s)  = 1  + (0.502 + 0.256i)2-s + (0.884 + 0.466i)3-s + (−0.400 − 0.551i)4-s + (−0.352 + 0.935i)5-s + (0.325 + 0.460i)6-s + (−1.11 − 1.11i)7-s + (−0.148 − 0.937i)8-s + (0.565 + 0.824i)9-s + (−0.416 + 0.380i)10-s + (0.236 − 0.0767i)11-s + (−0.0973 − 0.674i)12-s + (0.259 + 0.509i)13-s + (−0.274 − 0.843i)14-s + (−0.747 + 0.663i)15-s + (−0.0453 + 0.139i)16-s + (−0.527 + 0.0835i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1/2),\ 0.909 - 0.415i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19163 + 0.259605i\)
\(L(\frac12)\) \(\approx\) \(1.19163 + 0.259605i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.53 - 0.807i)T \)
5 \( 1 + (0.787 - 2.09i)T \)
good2 \( 1 + (-0.710 - 0.362i)T + (1.17 + 1.61i)T^{2} \)
7 \( 1 + (2.94 + 2.94i)T + 7iT^{2} \)
11 \( 1 + (-0.782 + 0.254i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.935 - 1.83i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (2.17 - 0.344i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-2.41 + 3.31i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.77 - 3.48i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (0.829 - 0.602i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.51 - 2.55i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.558 + 0.284i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-9.52 - 3.09i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-5.56 + 5.56i)T - 43iT^{2} \)
47 \( 1 + (0.246 - 1.55i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (12.7 + 2.02i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-3.55 + 10.9i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.751 + 2.31i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-1.37 - 8.69i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (5.06 + 6.97i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (7.14 + 3.63i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (2.31 + 3.19i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.19 + 7.54i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-2.35 - 7.24i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (2.77 + 0.439i)T + (92.2 + 29.9i)T^{2} \)
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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

−14.38598548022753089377582344390, −13.85173545035612820280389488604, −13.02773440295881608844371021776, −11.03480606757352141899928435074, −10.03846805573489520971126943560, −9.234990819457508021141034178104, −7.39890005433687310535532227913, −6.41486445442909842895416060528, −4.33459251989417272903272342821, −3.35984297867470345279104396316, 2.78804540572714013345071627332, 4.13420250331676270449131069669, 5.93131415713098712534448085326, 7.82420265837562069228365130542, 8.772505432177826383140310054232, 9.519272237846957300471483601344, 11.84538506962962937842846242888, 12.61482993511634493896249564268, 13.06389476360094712635170128084, 14.21790285190546047344193237420

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