Properties

Label 2-1617-1.1-c3-0-127
Degree $2$
Conductor $1617$
Sign $-1$
Analytic cond. $95.4060$
Root an. cond. $9.76760$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.06·2-s − 3·3-s − 6.86·4-s + 3.08·5-s − 3.19·6-s − 15.8·8-s + 9·9-s + 3.29·10-s + 11·11-s + 20.5·12-s − 46.1·13-s − 9.25·15-s + 38.0·16-s − 8.33·17-s + 9.59·18-s + 117.·19-s − 21.1·20-s + 11.7·22-s + 22.7·23-s + 47.5·24-s − 115.·25-s − 49.2·26-s − 27·27-s − 87.7·29-s − 9.87·30-s − 94.8·31-s + 167.·32-s + ⋯
L(s)  = 1  + 0.376·2-s − 0.577·3-s − 0.857·4-s + 0.276·5-s − 0.217·6-s − 0.700·8-s + 0.333·9-s + 0.104·10-s + 0.301·11-s + 0.495·12-s − 0.985·13-s − 0.159·15-s + 0.593·16-s − 0.118·17-s + 0.125·18-s + 1.41·19-s − 0.236·20-s + 0.113·22-s + 0.206·23-s + 0.404·24-s − 0.923·25-s − 0.371·26-s − 0.192·27-s − 0.562·29-s − 0.0600·30-s − 0.549·31-s + 0.924·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(95.4060\)
Root analytic conductor: \(9.76760\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1617,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
7 \( 1 \)
11 \( 1 - 11T \)
good2 \( 1 - 1.06T + 8T^{2} \)
5 \( 1 - 3.08T + 125T^{2} \)
13 \( 1 + 46.1T + 2.19e3T^{2} \)
17 \( 1 + 8.33T + 4.91e3T^{2} \)
19 \( 1 - 117.T + 6.85e3T^{2} \)
23 \( 1 - 22.7T + 1.21e4T^{2} \)
29 \( 1 + 87.7T + 2.43e4T^{2} \)
31 \( 1 + 94.8T + 2.97e4T^{2} \)
37 \( 1 - 86.0T + 5.06e4T^{2} \)
41 \( 1 - 223.T + 6.89e4T^{2} \)
43 \( 1 - 255.T + 7.95e4T^{2} \)
47 \( 1 - 168.T + 1.03e5T^{2} \)
53 \( 1 - 326.T + 1.48e5T^{2} \)
59 \( 1 - 106.T + 2.05e5T^{2} \)
61 \( 1 + 45.3T + 2.26e5T^{2} \)
67 \( 1 - 618.T + 3.00e5T^{2} \)
71 \( 1 - 91.0T + 3.57e5T^{2} \)
73 \( 1 - 277.T + 3.89e5T^{2} \)
79 \( 1 + 1.19e3T + 4.93e5T^{2} \)
83 \( 1 + 917.T + 5.71e5T^{2} \)
89 \( 1 + 1.61e3T + 7.04e5T^{2} \)
97 \( 1 + 969.T + 9.12e5T^{2} \)
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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

−8.796388527503121968550508475598, −7.69834641259301771366842413877, −7.04278887758298857174729082622, −5.78709593673066994941377088781, −5.46748058912001396567043932864, −4.50607359264678967302422481874, −3.74497779286445188678228495353, −2.55713083323267409150764835512, −1.12039988629188490211482412975, 0, 1.12039988629188490211482412975, 2.55713083323267409150764835512, 3.74497779286445188678228495353, 4.50607359264678967302422481874, 5.46748058912001396567043932864, 5.78709593673066994941377088781, 7.04278887758298857174729082622, 7.69834641259301771366842413877, 8.796388527503121968550508475598

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