Properties

Label 2-7350-1.1-c1-0-99
Degree $2$
Conductor $7350$
Sign $-1$
Analytic cond. $58.6900$
Root an. cond. $7.66094$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 3.05·11-s − 12-s − 1.64·13-s + 16-s + 2.90·17-s + 18-s − 2.21·19-s − 3.05·22-s + 1.78·23-s − 24-s − 1.64·26-s − 27-s + 5.58·29-s − 0.944·31-s + 32-s + 3.05·33-s + 2.90·34-s + 36-s − 10.7·37-s − 2.21·38-s + 1.64·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.921·11-s − 0.288·12-s − 0.455·13-s + 0.250·16-s + 0.705·17-s + 0.235·18-s − 0.507·19-s − 0.651·22-s + 0.373·23-s − 0.204·24-s − 0.321·26-s − 0.192·27-s + 1.03·29-s − 0.169·31-s + 0.176·32-s + 0.531·33-s + 0.498·34-s + 0.166·36-s − 1.76·37-s − 0.358·38-s + 0.262·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(58.6900\)
Root analytic conductor: \(7.66094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7350,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 3.05T + 11T^{2} \)
13 \( 1 + 1.64T + 13T^{2} \)
17 \( 1 - 2.90T + 17T^{2} \)
19 \( 1 + 2.21T + 19T^{2} \)
23 \( 1 - 1.78T + 23T^{2} \)
29 \( 1 - 5.58T + 29T^{2} \)
31 \( 1 + 0.944T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + 6.67T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 - 5.78T + 53T^{2} \)
59 \( 1 - 5.97T + 59T^{2} \)
61 \( 1 + 0.445T + 61T^{2} \)
67 \( 1 - 1.26T + 67T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 + 7.91T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + 0.874T + 83T^{2} \)
89 \( 1 + 17.5T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−7.33907606342810705927197868836, −6.81752162901963732511539260024, −6.03871400936173994315967475139, −5.28776704716552498925970173015, −4.93538072176081317361978832660, −4.05652292991667983012299092060, −3.17494825253390800553898680160, −2.41291603631733658802479833019, −1.35523795673424963995086121854, 0, 1.35523795673424963995086121854, 2.41291603631733658802479833019, 3.17494825253390800553898680160, 4.05652292991667983012299092060, 4.93538072176081317361978832660, 5.28776704716552498925970173015, 6.03871400936173994315967475139, 6.81752162901963732511539260024, 7.33907606342810705927197868836

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