Properties

Label 2-3675-1.1-c1-0-116
Degree $2$
Conductor $3675$
Sign $-1$
Analytic cond. $29.3450$
Root an. cond. $5.41710$
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  + 1.90·2-s − 3-s + 1.62·4-s − 1.90·6-s − 0.719·8-s + 9-s + 2·11-s − 1.62·12-s − 6.42·13-s − 4.61·16-s + 4.42·17-s + 1.90·18-s + 2.42·19-s + 3.80·22-s − 1.37·23-s + 0.719·24-s − 12.2·26-s − 27-s + 0.755·29-s − 5.18·31-s − 7.34·32-s − 2·33-s + 8.42·34-s + 1.62·36-s − 7.61·37-s + 4.62·38-s + 6.42·39-s + ⋯
L(s)  = 1  + 1.34·2-s − 0.577·3-s + 0.811·4-s − 0.776·6-s − 0.254·8-s + 0.333·9-s + 0.603·11-s − 0.468·12-s − 1.78·13-s − 1.15·16-s + 1.07·17-s + 0.448·18-s + 0.557·19-s + 0.811·22-s − 0.287·23-s + 0.146·24-s − 2.39·26-s − 0.192·27-s + 0.140·29-s − 0.931·31-s − 1.29·32-s − 0.348·33-s + 1.44·34-s + 0.270·36-s − 1.25·37-s + 0.749·38-s + 1.02·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(29.3450\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3675,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 - 1.90T + 2T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 6.42T + 13T^{2} \)
17 \( 1 - 4.42T + 17T^{2} \)
19 \( 1 - 2.42T + 19T^{2} \)
23 \( 1 + 1.37T + 23T^{2} \)
29 \( 1 - 0.755T + 29T^{2} \)
31 \( 1 + 5.18T + 31T^{2} \)
37 \( 1 + 7.61T + 37T^{2} \)
41 \( 1 - 8.23T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 2.75T + 47T^{2} \)
53 \( 1 + 9.18T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 + 6.85T + 61T^{2} \)
67 \( 1 + 2.75T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 1.57T + 73T^{2} \)
79 \( 1 + 4.85T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + 4.62T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−7.81846677711049997898068960152, −7.21539296370568739724433809459, −6.45840492919719691405433746095, −5.67746706699558493605285790778, −5.09155416546173539219780430690, −4.51890590672999027272270943376, −3.58698898384042368185291722638, −2.84821970732624238087221305756, −1.65825899413591127426315052974, 0, 1.65825899413591127426315052974, 2.84821970732624238087221305756, 3.58698898384042368185291722638, 4.51890590672999027272270943376, 5.09155416546173539219780430690, 5.67746706699558493605285790778, 6.45840492919719691405433746095, 7.21539296370568739724433809459, 7.81846677711049997898068960152

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