Properties

Label 2-35-1.1-c5-0-9
Degree $2$
Conductor $35$
Sign $-1$
Analytic cond. $5.61343$
Root an. cond. $2.36926$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.53·2-s − 10.5·3-s − 11.4·4-s − 25·5-s − 47.9·6-s − 49·7-s − 196.·8-s − 130.·9-s − 113.·10-s + 90.5·11-s + 121.·12-s − 74.8·13-s − 222.·14-s + 264.·15-s − 525.·16-s + 1.03e3·17-s − 592.·18-s − 31.6·19-s + 286.·20-s + 519.·21-s + 410.·22-s − 3.85e3·23-s + 2.08e3·24-s + 625·25-s − 339.·26-s + 3.95e3·27-s + 561.·28-s + ⋯
L(s)  = 1  + 0.800·2-s − 0.679·3-s − 0.358·4-s − 0.447·5-s − 0.544·6-s − 0.377·7-s − 1.08·8-s − 0.538·9-s − 0.358·10-s + 0.225·11-s + 0.243·12-s − 0.122·13-s − 0.302·14-s + 0.303·15-s − 0.513·16-s + 0.866·17-s − 0.431·18-s − 0.0201·19-s + 0.160·20-s + 0.256·21-s + 0.180·22-s − 1.52·23-s + 0.739·24-s + 0.200·25-s − 0.0983·26-s + 1.04·27-s + 0.135·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(5.61343\)
Root analytic conductor: \(2.36926\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 35,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 25T \)
7 \( 1 + 49T \)
good2 \( 1 - 4.53T + 32T^{2} \)
3 \( 1 + 10.5T + 243T^{2} \)
11 \( 1 - 90.5T + 1.61e5T^{2} \)
13 \( 1 + 74.8T + 3.71e5T^{2} \)
17 \( 1 - 1.03e3T + 1.41e6T^{2} \)
19 \( 1 + 31.6T + 2.47e6T^{2} \)
23 \( 1 + 3.85e3T + 6.43e6T^{2} \)
29 \( 1 + 866.T + 2.05e7T^{2} \)
31 \( 1 - 3.52e3T + 2.86e7T^{2} \)
37 \( 1 + 9.53e3T + 6.93e7T^{2} \)
41 \( 1 + 1.45e4T + 1.15e8T^{2} \)
43 \( 1 + 9.84e3T + 1.47e8T^{2} \)
47 \( 1 + 1.69e4T + 2.29e8T^{2} \)
53 \( 1 + 2.96e4T + 4.18e8T^{2} \)
59 \( 1 - 5.06e4T + 7.14e8T^{2} \)
61 \( 1 + 2.92e3T + 8.44e8T^{2} \)
67 \( 1 + 4.10e4T + 1.35e9T^{2} \)
71 \( 1 - 6.17e4T + 1.80e9T^{2} \)
73 \( 1 + 2.36e4T + 2.07e9T^{2} \)
79 \( 1 - 4.51e4T + 3.07e9T^{2} \)
83 \( 1 - 3.90e4T + 3.93e9T^{2} \)
89 \( 1 + 4.18e4T + 5.58e9T^{2} \)
97 \( 1 - 8.03e3T + 8.58e9T^{2} \)
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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

−14.73441856505347053689854138156, −13.74170258001829973367498041817, −12.35807842413409799800317033984, −11.67654231930691529753016475398, −9.975824195018469546036007313656, −8.365465018255132161276631766095, −6.34587109037505171982306508693, −5.10370852580996760227812753283, −3.48907564507549800331574950178, 0, 3.48907564507549800331574950178, 5.10370852580996760227812753283, 6.34587109037505171982306508693, 8.365465018255132161276631766095, 9.975824195018469546036007313656, 11.67654231930691529753016475398, 12.35807842413409799800317033984, 13.74170258001829973367498041817, 14.73441856505347053689854138156

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