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Properties

Label 2-75-3.2-c8-0-44
Degree $2$
Conductor $75$
Sign $-0.713 + 0.700i$
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  + 10.2i·2-s + (57.8 − 56.7i)3-s + 151.·4-s + (580. + 591. i)6-s − 3.44e3·7-s + 4.16e3i·8-s + (122. − 6.55e3i)9-s − 7.12e3i·11-s + (8.74e3 − 8.58e3i)12-s − 4.13e4·13-s − 3.52e4i·14-s − 3.95e3·16-s − 1.19e5i·17-s + (6.71e4 + 1.25e3i)18-s − 8.62e4·19-s + ⋯
L(s)  = 1  + 0.639i·2-s + (0.713 − 0.700i)3-s + 0.590·4-s + (0.448 + 0.456i)6-s − 1.43·7-s + 1.01i·8-s + (0.0187 − 0.999i)9-s − 0.486i·11-s + (0.421 − 0.413i)12-s − 1.44·13-s − 0.918i·14-s − 0.0602·16-s − 1.43i·17-s + (0.639 + 0.0119i)18-s − 0.662·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.713 + 0.700i$
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: \((2,\ 75,\ (\ :4),\ -0.713 + 0.700i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.300511 - 0.735222i\)
\(L(\frac12)\) \(\approx\) \(0.300511 - 0.735222i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-57.8 + 56.7i)T \)
5 \( 1 \)
good2 \( 1 - 10.2iT - 256T^{2} \)
7 \( 1 + 3.44e3T + 5.76e6T^{2} \)
11 \( 1 + 7.12e3iT - 2.14e8T^{2} \)
13 \( 1 + 4.13e4T + 8.15e8T^{2} \)
17 \( 1 + 1.19e5iT - 6.97e9T^{2} \)
19 \( 1 + 8.62e4T + 1.69e10T^{2} \)
23 \( 1 - 3.17e5iT - 7.83e10T^{2} \)
29 \( 1 - 5.98e4iT - 5.00e11T^{2} \)
31 \( 1 + 1.02e6T + 8.52e11T^{2} \)
37 \( 1 + 8.77e5T + 3.51e12T^{2} \)
41 \( 1 + 1.55e6iT - 7.98e12T^{2} \)
43 \( 1 - 2.56e6T + 1.16e13T^{2} \)
47 \( 1 + 8.98e6iT - 2.38e13T^{2} \)
53 \( 1 + 6.22e6iT - 6.22e13T^{2} \)
59 \( 1 - 3.96e6iT - 1.46e14T^{2} \)
61 \( 1 + 5.63e5T + 1.91e14T^{2} \)
67 \( 1 + 1.34e7T + 4.06e14T^{2} \)
71 \( 1 - 3.56e7iT - 6.45e14T^{2} \)
73 \( 1 + 9.70e5T + 8.06e14T^{2} \)
79 \( 1 + 2.78e7T + 1.51e15T^{2} \)
83 \( 1 - 2.25e7iT - 2.25e15T^{2} \)
89 \( 1 + 5.73e7iT - 3.93e15T^{2} \)
97 \( 1 - 3.31e7T + 7.83e15T^{2} \)
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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

−12.59836158429172254466159847651, −11.63882764915298739691054030529, −9.924407136450792813581340373909, −8.889112745735767030790321604088, −7.37943975997772409866662254289, −6.90698697960254655580252089934, −5.60556625272282818787095759042, −3.27364243305551882032655978928, −2.25589152130606227332873568252, −0.19218995949751685574584437598, 2.10266691145021184419945381529, 3.08331635674378955242423175024, 4.28743144683405686022613079323, 6.27962770388519714905746229436, 7.53079932235833305978574030686, 9.176691033570218360022142408972, 10.07120769798718235462479471083, 10.71751929404770877174285497340, 12.46555658731027006645169928808, 12.86174961172690960628521252718

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