Properties

Label 2-35-1.1-c3-0-4
Degree $2$
Conductor $35$
Sign $1$
Analytic cond. $2.06506$
Root an. cond. $1.43703$
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  + 5.41·2-s − 4.65·3-s + 21.3·4-s − 5·5-s − 25.2·6-s − 7·7-s + 72.0·8-s − 5.31·9-s − 27.0·10-s − 52.2·11-s − 99.2·12-s + 30.6·13-s − 37.8·14-s + 23.2·15-s + 219.·16-s + 37.2·17-s − 28.7·18-s + 80.2·19-s − 106.·20-s + 32.5·21-s − 282.·22-s + 25.8·23-s − 335.·24-s + 25·25-s + 165.·26-s + 150.·27-s − 149.·28-s + ⋯
L(s)  = 1  + 1.91·2-s − 0.896·3-s + 2.66·4-s − 0.447·5-s − 1.71·6-s − 0.377·7-s + 3.18·8-s − 0.196·9-s − 0.856·10-s − 1.43·11-s − 2.38·12-s + 0.654·13-s − 0.723·14-s + 0.400·15-s + 3.43·16-s + 0.531·17-s − 0.376·18-s + 0.968·19-s − 1.19·20-s + 0.338·21-s − 2.74·22-s + 0.234·23-s − 2.85·24-s + 0.200·25-s + 1.25·26-s + 1.07·27-s − 1.00·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $1$
Analytic conductor: \(2.06506\)
Root analytic conductor: \(1.43703\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.430073287\)
\(L(\frac12)\) \(\approx\) \(2.430073287\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5T \)
7 \( 1 + 7T \)
good2 \( 1 - 5.41T + 8T^{2} \)
3 \( 1 + 4.65T + 27T^{2} \)
11 \( 1 + 52.2T + 1.33e3T^{2} \)
13 \( 1 - 30.6T + 2.19e3T^{2} \)
17 \( 1 - 37.2T + 4.91e3T^{2} \)
19 \( 1 - 80.2T + 6.85e3T^{2} \)
23 \( 1 - 25.8T + 1.21e4T^{2} \)
29 \( 1 - 20.9T + 2.43e4T^{2} \)
31 \( 1 + 314.T + 2.97e4T^{2} \)
37 \( 1 - 197.T + 5.06e4T^{2} \)
41 \( 1 - 11.3T + 6.89e4T^{2} \)
43 \( 1 + 33.8T + 7.95e4T^{2} \)
47 \( 1 + 361.T + 1.03e5T^{2} \)
53 \( 1 - 153.T + 1.48e5T^{2} \)
59 \( 1 + 616T + 2.05e5T^{2} \)
61 \( 1 - 15.2T + 2.26e5T^{2} \)
67 \( 1 + 166.T + 3.00e5T^{2} \)
71 \( 1 + 952T + 3.57e5T^{2} \)
73 \( 1 + 148.T + 3.89e5T^{2} \)
79 \( 1 - 857.T + 4.93e5T^{2} \)
83 \( 1 - 660.T + 5.71e5T^{2} \)
89 \( 1 + 45.7T + 7.04e5T^{2} \)
97 \( 1 - 1.68e3T + 9.12e5T^{2} \)
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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

−15.96937458453203323925251089762, −14.78087386236095405578224077907, −13.46188577452627963038114562248, −12.55080475015299726901144508149, −11.49866612153758956587525004771, −10.63887138374207418976842958442, −7.52985839123901122492131590224, −5.99925029535903665821127577129, −5.05268532606638870053803240027, −3.21748054548271073994916860722, 3.21748054548271073994916860722, 5.05268532606638870053803240027, 5.99925029535903665821127577129, 7.52985839123901122492131590224, 10.63887138374207418976842958442, 11.49866612153758956587525004771, 12.55080475015299726901144508149, 13.46188577452627963038114562248, 14.78087386236095405578224077907, 15.96937458453203323925251089762

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