Properties

Label 2-35-1.1-c9-0-15
Degree $2$
Conductor $35$
Sign $-1$
Analytic cond. $18.0262$
Root an. cond. $4.24573$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 28·2-s − 116·3-s + 272·4-s + 625·5-s − 3.24e3·6-s + 2.40e3·7-s − 6.72e3·8-s − 6.22e3·9-s + 1.75e4·10-s − 2.55e4·11-s − 3.15e4·12-s − 4.23e4·13-s + 6.72e4·14-s − 7.25e4·15-s − 3.27e5·16-s − 5.26e5·17-s − 1.74e5·18-s − 3.50e5·19-s + 1.70e5·20-s − 2.78e5·21-s − 7.15e5·22-s − 6.21e5·23-s + 7.79e5·24-s + 3.90e5·25-s − 1.18e6·26-s + 3.00e6·27-s + 6.53e5·28-s + ⋯
L(s)  = 1  + 1.23·2-s − 0.826·3-s + 0.531·4-s + 0.447·5-s − 1.02·6-s + 0.377·7-s − 0.580·8-s − 0.316·9-s + 0.553·10-s − 0.526·11-s − 0.439·12-s − 0.410·13-s + 0.467·14-s − 0.369·15-s − 1.24·16-s − 1.52·17-s − 0.391·18-s − 0.616·19-s + 0.237·20-s − 0.312·21-s − 0.651·22-s − 0.463·23-s + 0.479·24-s + 1/5·25-s − 0.508·26-s + 1.08·27-s + 0.200·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(18.0262\)
Root analytic conductor: \(4.24573\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: $\chi_{35} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 35,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - p^{4} T \)
7 \( 1 - p^{4} T \)
good2 \( 1 - 7 p^{2} T + p^{9} T^{2} \)
3 \( 1 + 116 T + p^{9} T^{2} \)
11 \( 1 + 25548 T + p^{9} T^{2} \)
13 \( 1 + 42306 T + p^{9} T^{2} \)
17 \( 1 + 526342 T + p^{9} T^{2} \)
19 \( 1 + 350060 T + p^{9} T^{2} \)
23 \( 1 + 621976 T + p^{9} T^{2} \)
29 \( 1 - 6720430 T + p^{9} T^{2} \)
31 \( 1 + 6412208 T + p^{9} T^{2} \)
37 \( 1 + 2317682 T + p^{9} T^{2} \)
41 \( 1 + 10224678 T + p^{9} T^{2} \)
43 \( 1 - 30114004 T + p^{9} T^{2} \)
47 \( 1 + 23644912 T + p^{9} T^{2} \)
53 \( 1 - 57292654 T + p^{9} T^{2} \)
59 \( 1 - 84934780 T + p^{9} T^{2} \)
61 \( 1 - 14677822 T + p^{9} T^{2} \)
67 \( 1 + 244557812 T + p^{9} T^{2} \)
71 \( 1 - 61901952 T + p^{9} T^{2} \)
73 \( 1 + 283763726 T + p^{9} T^{2} \)
79 \( 1 - 276107480 T + p^{9} T^{2} \)
83 \( 1 + 72995956 T + p^{9} T^{2} \)
89 \( 1 + 896368470 T + p^{9} T^{2} \)
97 \( 1 - 1205809578 T + p^{9} T^{2} \)
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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

−13.82706994463753367755988528351, −12.77780797516303524791171987966, −11.71522569981137308570261655711, −10.60093104081596524156837200727, −8.757475580146295307777216922596, −6.60214279484499050801103175578, −5.49379448429270748142859265401, −4.47552726515895085107292806570, −2.48792095993394324947306954592, 0, 2.48792095993394324947306954592, 4.47552726515895085107292806570, 5.49379448429270748142859265401, 6.60214279484499050801103175578, 8.757475580146295307777216922596, 10.60093104081596524156837200727, 11.71522569981137308570261655711, 12.77780797516303524791171987966, 13.82706994463753367755988528351

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