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Properties

Label 2-75-5.3-c8-0-8
Degree $2$
Conductor $75$
Sign $0.229 - 0.973i$
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  + (−6.62 + 6.62i)2-s + (−33.0 − 33.0i)3-s + 168. i·4-s + 437.·6-s + (216. − 216. i)7-s + (−2.80e3 − 2.80e3i)8-s + 2.18e3i·9-s + 1.33e4·11-s + (5.56e3 − 5.56e3i)12-s + (−3.69e3 − 3.69e3i)13-s + 2.86e3i·14-s − 5.89e3·16-s + (6.02e3 − 6.02e3i)17-s + (−1.44e4 − 1.44e4i)18-s − 2.01e5i·19-s + ⋯
L(s)  = 1  + (−0.413 + 0.413i)2-s + (−0.408 − 0.408i)3-s + 0.657i·4-s + 0.337·6-s + (0.0902 − 0.0902i)7-s + (−0.685 − 0.685i)8-s + 0.333i·9-s + 0.911·11-s + (0.268 − 0.268i)12-s + (−0.129 − 0.129i)13-s + 0.0746i·14-s − 0.0900·16-s + (0.0721 − 0.0721i)17-s + (−0.137 − 0.137i)18-s − 1.54i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.930741 + 0.736605i\)
\(L(\frac12)\) \(\approx\) \(0.930741 + 0.736605i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (33.0 + 33.0i)T \)
5 \( 1 \)
good2 \( 1 + (6.62 - 6.62i)T - 256iT^{2} \)
7 \( 1 + (-216. + 216. i)T - 5.76e6iT^{2} \)
11 \( 1 - 1.33e4T + 2.14e8T^{2} \)
13 \( 1 + (3.69e3 + 3.69e3i)T + 8.15e8iT^{2} \)
17 \( 1 + (-6.02e3 + 6.02e3i)T - 6.97e9iT^{2} \)
19 \( 1 + 2.01e5iT - 1.69e10T^{2} \)
23 \( 1 + (8.58e3 + 8.58e3i)T + 7.83e10iT^{2} \)
29 \( 1 - 1.18e6iT - 5.00e11T^{2} \)
31 \( 1 - 1.38e6T + 8.52e11T^{2} \)
37 \( 1 + (1.93e6 - 1.93e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 1.77e6T + 7.98e12T^{2} \)
43 \( 1 + (-2.97e6 - 2.97e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (5.71e6 - 5.71e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (-1.32e6 - 1.32e6i)T + 6.22e13iT^{2} \)
59 \( 1 - 1.28e7iT - 1.46e14T^{2} \)
61 \( 1 - 8.32e6T + 1.91e14T^{2} \)
67 \( 1 + (-1.90e7 + 1.90e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 9.37e6T + 6.45e14T^{2} \)
73 \( 1 + (-3.23e7 - 3.23e7i)T + 8.06e14iT^{2} \)
79 \( 1 + 3.25e7iT - 1.51e15T^{2} \)
83 \( 1 + (1.53e7 + 1.53e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 1.68e7iT - 3.93e15T^{2} \)
97 \( 1 + (-4.53e7 + 4.53e7i)T - 7.83e15iT^{2} \)
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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

−12.95399290581460250117640373901, −12.05941260024170492974014866415, −11.06958583334303546460543787394, −9.468096807125003498199241509847, −8.440719528261177047947446484696, −7.21170291789081439235213816019, −6.40878739064554773603955020312, −4.62940354600289539358306461480, −2.96467616057248038185755829590, −0.976943798450479862735005476085, 0.58695911739054447296734030522, 1.97845365727018364701732730936, 3.93344821136667607277747803446, 5.45258337295305100330272654364, 6.48766415448950820873003836119, 8.344326975560099127978234202813, 9.585491201044538893199447595916, 10.27792818544910311193160865337, 11.47046456436702372529548038484, 12.18368395741917853309432125675

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