Properties

Label 2-675-1.1-c3-0-11
Degree $2$
Conductor $675$
Sign $1$
Analytic cond. $39.8262$
Root an. cond. $6.31080$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.12·2-s − 3.47·4-s − 30.7·7-s − 24.4·8-s − 50.1·11-s + 15.9·13-s − 65.2·14-s − 24.0·16-s + 105.·17-s − 21.3·19-s − 106.·22-s + 136.·23-s + 33.9·26-s + 106.·28-s + 224.·29-s − 225.·31-s + 144.·32-s + 224.·34-s + 416.·37-s − 45.2·38-s − 76.1·41-s − 31.7·43-s + 174.·44-s + 289.·46-s + 60.8·47-s + 599.·49-s − 55.5·52-s + ⋯
L(s)  = 1  + 0.751·2-s − 0.434·4-s − 1.65·7-s − 1.07·8-s − 1.37·11-s + 0.340·13-s − 1.24·14-s − 0.375·16-s + 1.50·17-s − 0.257·19-s − 1.03·22-s + 1.23·23-s + 0.255·26-s + 0.720·28-s + 1.43·29-s − 1.30·31-s + 0.796·32-s + 1.13·34-s + 1.85·37-s − 0.193·38-s − 0.290·41-s − 0.112·43-s + 0.597·44-s + 0.927·46-s + 0.188·47-s + 1.74·49-s − 0.148·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(39.8262\)
Root analytic conductor: \(6.31080\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.418824954\)
\(L(\frac12)\) \(\approx\) \(1.418824954\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 - 2.12T + 8T^{2} \)
7 \( 1 + 30.7T + 343T^{2} \)
11 \( 1 + 50.1T + 1.33e3T^{2} \)
13 \( 1 - 15.9T + 2.19e3T^{2} \)
17 \( 1 - 105.T + 4.91e3T^{2} \)
19 \( 1 + 21.3T + 6.85e3T^{2} \)
23 \( 1 - 136.T + 1.21e4T^{2} \)
29 \( 1 - 224.T + 2.43e4T^{2} \)
31 \( 1 + 225.T + 2.97e4T^{2} \)
37 \( 1 - 416.T + 5.06e4T^{2} \)
41 \( 1 + 76.1T + 6.89e4T^{2} \)
43 \( 1 + 31.7T + 7.95e4T^{2} \)
47 \( 1 - 60.8T + 1.03e5T^{2} \)
53 \( 1 + 466.T + 1.48e5T^{2} \)
59 \( 1 + 95.4T + 2.05e5T^{2} \)
61 \( 1 + 357.T + 2.26e5T^{2} \)
67 \( 1 + 87.8T + 3.00e5T^{2} \)
71 \( 1 + 412.T + 3.57e5T^{2} \)
73 \( 1 - 331.T + 3.89e5T^{2} \)
79 \( 1 + 248.T + 4.93e5T^{2} \)
83 \( 1 + 552.T + 5.71e5T^{2} \)
89 \( 1 - 291.T + 7.04e5T^{2} \)
97 \( 1 + 198.T + 9.12e5T^{2} \)
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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

−9.986072351732090829283301485424, −9.412203176294440035645715537641, −8.412739929674706111671305661728, −7.36218483126207473098929166947, −6.22547405904543831979635479420, −5.59337233525919254444864782976, −4.60048211768182198913934116324, −3.30822491891973609950467231870, −2.90782732048414088117121038546, −0.59800648615558948276356389523, 0.59800648615558948276356389523, 2.90782732048414088117121038546, 3.30822491891973609950467231870, 4.60048211768182198913934116324, 5.59337233525919254444864782976, 6.22547405904543831979635479420, 7.36218483126207473098929166947, 8.412739929674706111671305661728, 9.412203176294440035645715537641, 9.986072351732090829283301485424

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