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Properties

Label 2-75-5.3-c8-0-23
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  + (21.2 − 21.2i)2-s + (−33.0 − 33.0i)3-s − 646. i·4-s − 1.40e3·6-s + (−1.86e3 + 1.86e3i)7-s + (−8.30e3 − 8.30e3i)8-s + 2.18e3i·9-s − 3.92e3·11-s + (−2.13e4 + 2.13e4i)12-s + (3.47e3 + 3.47e3i)13-s + 7.91e4i·14-s − 1.87e5·16-s + (−1.00e5 + 1.00e5i)17-s + (4.64e4 + 4.64e4i)18-s − 1.52e5i·19-s + ⋯
L(s)  = 1  + (1.32 − 1.32i)2-s + (−0.408 − 0.408i)3-s − 2.52i·4-s − 1.08·6-s + (−0.775 + 0.775i)7-s + (−2.02 − 2.02i)8-s + 0.333i·9-s − 0.268·11-s + (−1.03 + 1.03i)12-s + (0.121 + 0.121i)13-s + 2.05i·14-s − 2.85·16-s + (−1.19 + 1.19i)17-s + (0.442 + 0.442i)18-s − 1.17i·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

\(L(\frac{9}{2})\) \(\approx\) \(0.572831 + 0.453349i\)
\(L(\frac12)\) \(\approx\) \(0.572831 + 0.453349i\)
\(L(5)\) 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 \( 1 + (-21.2 + 21.2i)T - 256iT^{2} \)
7 \( 1 + (1.86e3 - 1.86e3i)T - 5.76e6iT^{2} \)
11 \( 1 + 3.92e3T + 2.14e8T^{2} \)
13 \( 1 + (-3.47e3 - 3.47e3i)T + 8.15e8iT^{2} \)
17 \( 1 + (1.00e5 - 1.00e5i)T - 6.97e9iT^{2} \)
19 \( 1 + 1.52e5iT - 1.69e10T^{2} \)
23 \( 1 + (-7.61e4 - 7.61e4i)T + 7.83e10iT^{2} \)
29 \( 1 + 9.35e5iT - 5.00e11T^{2} \)
31 \( 1 + 2.73e5T + 8.52e11T^{2} \)
37 \( 1 + (1.05e6 - 1.05e6i)T - 3.51e12iT^{2} \)
41 \( 1 + 2.83e6T + 7.98e12T^{2} \)
43 \( 1 + (-3.82e5 - 3.82e5i)T + 1.16e13iT^{2} \)
47 \( 1 + (-2.79e6 + 2.79e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (-1.27e6 - 1.27e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 9.64e6iT - 1.46e14T^{2} \)
61 \( 1 + 8.78e6T + 1.91e14T^{2} \)
67 \( 1 + (1.74e7 - 1.74e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 2.53e7T + 6.45e14T^{2} \)
73 \( 1 + (3.10e7 + 3.10e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 3.27e7iT - 1.51e15T^{2} \)
83 \( 1 + (1.57e7 + 1.57e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 2.96e7iT - 3.93e15T^{2} \)
97 \( 1 + (-7.80e7 + 7.80e7i)T - 7.83e15iT^{2} \)
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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

−12.11295170771386182415191680181, −11.27191101564391502312949166834, −10.27703633744022436886972706986, −8.974063317853998428210779141475, −6.59331364586578804081468290819, −5.68432582033724600524127192689, −4.39845739792404751225871378458, −2.92539350923983815270546020753, −1.83884991834428741756918983659, −0.14128730337245341397700025757, 3.18492830077030678088419701222, 4.28110451837586345910194153076, 5.39175345950179268054391812848, 6.57606297147699175386768789885, 7.38591561253967356165794396437, 8.932939166970634234353174790098, 10.56519210129605769766080101565, 11.99355079757641038906162184034, 13.01108899057386802842336413772, 13.79661901430653293343860714638

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