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Properties

Label 2-75-5.2-c8-0-2
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  + (−4.09 − 4.09i)2-s + (33.0 − 33.0i)3-s − 222. i·4-s − 270.·6-s + (2.63e3 + 2.63e3i)7-s + (−1.95e3 + 1.95e3i)8-s − 2.18e3i·9-s − 2.16e4·11-s + (−7.35e3 − 7.35e3i)12-s + (−7.10e3 + 7.10e3i)13-s − 2.15e4i·14-s − 4.09e4·16-s + (−6.58e4 − 6.58e4i)17-s + (−8.94e3 + 8.94e3i)18-s + 1.58e5i·19-s + ⋯
L(s)  = 1  + (−0.255 − 0.255i)2-s + (0.408 − 0.408i)3-s − 0.869i·4-s − 0.208·6-s + (1.09 + 1.09i)7-s + (−0.477 + 0.477i)8-s − 0.333i·9-s − 1.47·11-s + (−0.354 − 0.354i)12-s + (−0.248 + 0.248i)13-s − 0.561i·14-s − 0.624·16-s + (−0.788 − 0.788i)17-s + (−0.0852 + 0.0852i)18-s + 1.21i·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: $\chi_{75} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.270002 + 0.341162i\)
\(L(\frac12)\) \(\approx\) \(0.270002 + 0.341162i\)
\(L(5)\) 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 \( 1 + (4.09 + 4.09i)T + 256iT^{2} \)
7 \( 1 + (-2.63e3 - 2.63e3i)T + 5.76e6iT^{2} \)
11 \( 1 + 2.16e4T + 2.14e8T^{2} \)
13 \( 1 + (7.10e3 - 7.10e3i)T - 8.15e8iT^{2} \)
17 \( 1 + (6.58e4 + 6.58e4i)T + 6.97e9iT^{2} \)
19 \( 1 - 1.58e5iT - 1.69e10T^{2} \)
23 \( 1 + (1.03e5 - 1.03e5i)T - 7.83e10iT^{2} \)
29 \( 1 + 3.14e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.54e6T + 8.52e11T^{2} \)
37 \( 1 + (-9.97e5 - 9.97e5i)T + 3.51e12iT^{2} \)
41 \( 1 - 6.20e5T + 7.98e12T^{2} \)
43 \( 1 + (3.53e6 - 3.53e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (-1.94e6 - 1.94e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (-1.25e5 + 1.25e5i)T - 6.22e13iT^{2} \)
59 \( 1 - 1.06e6iT - 1.46e14T^{2} \)
61 \( 1 - 7.07e6T + 1.91e14T^{2} \)
67 \( 1 + (-5.96e6 - 5.96e6i)T + 4.06e14iT^{2} \)
71 \( 1 + 3.43e7T + 6.45e14T^{2} \)
73 \( 1 + (-5.18e6 + 5.18e6i)T - 8.06e14iT^{2} \)
79 \( 1 + 1.54e7iT - 1.51e15T^{2} \)
83 \( 1 + (-2.37e7 + 2.37e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 3.64e7iT - 3.93e15T^{2} \)
97 \( 1 + (-4.94e7 - 4.94e7i)T + 7.83e15iT^{2} \)
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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

−13.24154397267982664014608283094, −11.91907842989431163898093842487, −11.03456382147284286862190053936, −9.771379292596710442578388575726, −8.666798066431380227927960579099, −7.66308159289680672689068121823, −5.88239611933059361537242808836, −4.93222085922268531905770577495, −2.52614518817072203920448790226, −1.68736533632798105529541182558, 0.12800033857200766367408182205, 2.32858790384632046115590901000, 3.85888014592302318251219321127, 4.99094805388269253515148323489, 7.15401076751930678008894112202, 7.930381792629587855925633058238, 8.850355162472079618824605295785, 10.43306709762148304496563594382, 11.18064314662303870756118586569, 12.82713573293304645034851566149

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