Properties

Label 2-17e2-1.1-c9-0-30
Degree $2$
Conductor $289$
Sign $1$
Analytic cond. $148.845$
Root an. cond. $12.2002$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 43.7·2-s + 83.5·3-s + 1.40e3·4-s − 712.·5-s − 3.65e3·6-s + 5.54e3·7-s − 3.89e4·8-s − 1.26e4·9-s + 3.11e4·10-s + 3.60e4·11-s + 1.17e5·12-s − 1.75e5·13-s − 2.42e5·14-s − 5.95e4·15-s + 9.84e5·16-s + 5.55e5·18-s − 5.12e5·19-s − 9.98e5·20-s + 4.63e5·21-s − 1.57e6·22-s − 1.41e6·23-s − 3.25e6·24-s − 1.44e6·25-s + 7.67e6·26-s − 2.70e6·27-s + 7.76e6·28-s − 6.23e6·29-s + ⋯
L(s)  = 1  − 1.93·2-s + 0.595·3-s + 2.73·4-s − 0.509·5-s − 1.15·6-s + 0.872·7-s − 3.35·8-s − 0.644·9-s + 0.985·10-s + 0.743·11-s + 1.63·12-s − 1.70·13-s − 1.68·14-s − 0.303·15-s + 3.75·16-s + 1.24·18-s − 0.902·19-s − 1.39·20-s + 0.519·21-s − 1.43·22-s − 1.05·23-s − 2.00·24-s − 0.740·25-s + 3.29·26-s − 0.980·27-s + 2.38·28-s − 1.63·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $1$
Analytic conductor: \(148.845\)
Root analytic conductor: \(12.2002\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.4099606389\)
\(L(\frac12)\) \(\approx\) \(0.4099606389\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + 43.7T + 512T^{2} \)
3 \( 1 - 83.5T + 1.96e4T^{2} \)
5 \( 1 + 712.T + 1.95e6T^{2} \)
7 \( 1 - 5.54e3T + 4.03e7T^{2} \)
11 \( 1 - 3.60e4T + 2.35e9T^{2} \)
13 \( 1 + 1.75e5T + 1.06e10T^{2} \)
19 \( 1 + 5.12e5T + 3.22e11T^{2} \)
23 \( 1 + 1.41e6T + 1.80e12T^{2} \)
29 \( 1 + 6.23e6T + 1.45e13T^{2} \)
31 \( 1 + 1.00e5T + 2.64e13T^{2} \)
37 \( 1 + 1.77e7T + 1.29e14T^{2} \)
41 \( 1 - 2.70e7T + 3.27e14T^{2} \)
43 \( 1 - 7.78e5T + 5.02e14T^{2} \)
47 \( 1 - 1.03e7T + 1.11e15T^{2} \)
53 \( 1 - 3.35e7T + 3.29e15T^{2} \)
59 \( 1 - 1.32e8T + 8.66e15T^{2} \)
61 \( 1 + 4.00e7T + 1.16e16T^{2} \)
67 \( 1 - 2.73e8T + 2.72e16T^{2} \)
71 \( 1 - 1.50e8T + 4.58e16T^{2} \)
73 \( 1 - 1.55e7T + 5.88e16T^{2} \)
79 \( 1 + 4.95e7T + 1.19e17T^{2} \)
83 \( 1 + 2.23e8T + 1.86e17T^{2} \)
89 \( 1 - 3.53e7T + 3.50e17T^{2} \)
97 \( 1 + 1.41e9T + 7.60e17T^{2} \)
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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

−9.902913627961056010592479407476, −9.193322419161827540475130111076, −8.328270908134123265542750990819, −7.76140548563047452515989491854, −6.98324611532033645284866585113, −5.63648315845313463712632099230, −3.83693662208917860487103373737, −2.37209993878298944866706660632, −1.84967177180516146745348651168, −0.34964157972179207476644448935, 0.34964157972179207476644448935, 1.84967177180516146745348651168, 2.37209993878298944866706660632, 3.83693662208917860487103373737, 5.63648315845313463712632099230, 6.98324611532033645284866585113, 7.76140548563047452515989491854, 8.328270908134123265542750990819, 9.193322419161827540475130111076, 9.902913627961056010592479407476

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