Properties

Label 2-75-75.8-c1-0-1
Degree $2$
Conductor $75$
Sign $0.444 - 0.895i$
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  + (0.0231 − 0.0455i)2-s + (−1.39 + 1.03i)3-s + (1.17 + 1.61i)4-s + (−1.22 + 1.86i)5-s + (0.0147 + 0.0872i)6-s + (1.44 − 1.44i)7-s + (0.201 − 0.0319i)8-s + (0.866 − 2.87i)9-s + (0.0565 + 0.0992i)10-s + (2.86 − 0.931i)11-s + (−3.30 − 1.03i)12-s + (−2.79 + 1.42i)13-s + (−0.0321 − 0.0990i)14-s + (−0.222 − 3.86i)15-s + (−1.23 + 3.78i)16-s + (−0.952 − 6.01i)17-s + ⋯
L(s)  = 1  + (0.0164 − 0.0321i)2-s + (−0.802 + 0.596i)3-s + (0.587 + 0.807i)4-s + (−0.549 + 0.835i)5-s + (0.00603 + 0.0356i)6-s + (0.544 − 0.544i)7-s + (0.0713 − 0.0112i)8-s + (0.288 − 0.957i)9-s + (0.0178 + 0.0313i)10-s + (0.864 − 0.280i)11-s + (−0.953 − 0.298i)12-s + (−0.774 + 0.394i)13-s + (−0.00860 − 0.0264i)14-s + (−0.0575 − 0.998i)15-s + (−0.307 + 0.947i)16-s + (−0.230 − 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1/2),\ 0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.701023 + 0.434527i\)
\(L(\frac12)\) \(\approx\) \(0.701023 + 0.434527i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.39 - 1.03i)T \)
5 \( 1 + (1.22 - 1.86i)T \)
good2 \( 1 + (-0.0231 + 0.0455i)T + (-1.17 - 1.61i)T^{2} \)
7 \( 1 + (-1.44 + 1.44i)T - 7iT^{2} \)
11 \( 1 + (-2.86 + 0.931i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (2.79 - 1.42i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.952 + 6.01i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (-1.91 + 2.63i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-5.12 - 2.61i)T + (13.5 + 18.6i)T^{2} \)
29 \( 1 + (0.976 - 0.709i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.0161 + 0.0117i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.24 - 2.44i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (10.1 + 3.31i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (3.91 + 3.91i)T + 43iT^{2} \)
47 \( 1 + (-3.63 - 0.575i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (0.635 - 4.01i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (1.33 - 4.11i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.99 + 12.2i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (0.920 - 0.145i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (5.79 + 7.97i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.0177 + 0.0348i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (6.00 + 8.26i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.17 - 0.344i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (-4.18 - 12.8i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.838 - 5.29i)T + (-92.2 - 29.9i)T^{2} \)
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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

−14.97907639368613822221055344200, −13.75714875898458614391143358329, −11.96419573440297890602896074654, −11.54556600453923943854640897910, −10.70466356843379253623445729923, −9.212120303866334535171327442703, −7.38403036126770512370553394120, −6.75445950186798943007438946079, −4.69654225408392204704756873490, −3.30404473757755661058349096788, 1.57733821693861113494993648667, 4.78687672861061233630953616521, 5.85760982577749677870353602942, 7.17000631682365384764986938973, 8.487589285646723663970660223439, 10.09347772344143301595924523782, 11.35007667737477342838473534436, 12.04050969389297622352535964020, 12.95682351786501531026127546487, 14.61695190470020273897204874462

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