Properties

Label 2-75-1.1-c21-0-12
Degree $2$
Conductor $75$
Sign $1$
Analytic cond. $209.608$
Root an. cond. $14.4778$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 359.·2-s − 5.90e4·3-s − 1.96e6·4-s − 2.12e7·6-s − 2.25e8·7-s − 1.46e9·8-s + 3.48e9·9-s − 3.39e10·11-s + 1.16e11·12-s + 7.87e11·13-s − 8.12e10·14-s + 3.59e12·16-s − 1.90e12·17-s + 1.25e12·18-s + 3.53e13·19-s + 1.33e13·21-s − 1.22e13·22-s − 1.05e14·23-s + 8.63e13·24-s + 2.83e14·26-s − 2.05e14·27-s + 4.44e14·28-s − 1.97e15·29-s − 6.94e15·31-s + 4.36e15·32-s + 2.00e15·33-s − 6.85e14·34-s + ⋯
L(s)  = 1  + 0.248·2-s − 0.577·3-s − 0.938·4-s − 0.143·6-s − 0.302·7-s − 0.481·8-s + 0.333·9-s − 0.394·11-s + 0.541·12-s + 1.58·13-s − 0.0750·14-s + 0.818·16-s − 0.229·17-s + 0.0828·18-s + 1.32·19-s + 0.174·21-s − 0.0980·22-s − 0.528·23-s + 0.278·24-s + 0.393·26-s − 0.192·27-s + 0.283·28-s − 0.872·29-s − 1.52·31-s + 0.685·32-s + 0.227·33-s − 0.0569·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(209.608\)
Root analytic conductor: \(14.4778\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(1.118834263\)
\(L(\frac12)\) \(\approx\) \(1.118834263\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.90e4T \)
5 \( 1 \)
good2 \( 1 - 359.T + 2.09e6T^{2} \)
7 \( 1 + 2.25e8T + 5.58e17T^{2} \)
11 \( 1 + 3.39e10T + 7.40e21T^{2} \)
13 \( 1 - 7.87e11T + 2.47e23T^{2} \)
17 \( 1 + 1.90e12T + 6.90e25T^{2} \)
19 \( 1 - 3.53e13T + 7.14e26T^{2} \)
23 \( 1 + 1.05e14T + 3.94e28T^{2} \)
29 \( 1 + 1.97e15T + 5.13e30T^{2} \)
31 \( 1 + 6.94e15T + 2.08e31T^{2} \)
37 \( 1 + 3.71e16T + 8.55e32T^{2} \)
41 \( 1 - 1.35e17T + 7.38e33T^{2} \)
43 \( 1 - 4.58e16T + 2.00e34T^{2} \)
47 \( 1 + 4.29e17T + 1.30e35T^{2} \)
53 \( 1 - 2.26e18T + 1.62e36T^{2} \)
59 \( 1 - 2.27e17T + 1.54e37T^{2} \)
61 \( 1 - 3.52e18T + 3.10e37T^{2} \)
67 \( 1 + 1.45e19T + 2.22e38T^{2} \)
71 \( 1 + 2.96e19T + 7.52e38T^{2} \)
73 \( 1 + 3.47e19T + 1.34e39T^{2} \)
79 \( 1 - 3.47e19T + 7.08e39T^{2} \)
83 \( 1 + 1.56e20T + 1.99e40T^{2} \)
89 \( 1 - 1.89e20T + 8.65e40T^{2} \)
97 \( 1 - 1.11e20T + 5.27e41T^{2} \)
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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

−10.67265113791775606273882207917, −9.557076997750276294556611511994, −8.641437873935290443551302582823, −7.39277836795651454847016148497, −5.96764543020526554027583774164, −5.35154485112876716949522485371, −4.07264875770822596449299158016, −3.28761270976965617050578862185, −1.52539227348374934974886135493, −0.46221496347047401294696333995, 0.46221496347047401294696333995, 1.52539227348374934974886135493, 3.28761270976965617050578862185, 4.07264875770822596449299158016, 5.35154485112876716949522485371, 5.96764543020526554027583774164, 7.39277836795651454847016148497, 8.641437873935290443551302582823, 9.557076997750276294556611511994, 10.67265113791775606273882207917

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