Properties

Label 2-35-1.1-c5-0-4
Degree $2$
Conductor $35$
Sign $-1$
Analytic cond. $5.61343$
Root an. cond. $2.36926$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 3-s + 32·4-s + 25·5-s − 8·6-s + 49·7-s − 242·9-s − 200·10-s − 453·11-s + 32·12-s − 969·13-s − 392·14-s + 25·15-s − 1.02e3·16-s + 1.63e3·17-s + 1.93e3·18-s − 1.55e3·19-s + 800·20-s + 49·21-s + 3.62e3·22-s − 1.65e3·23-s + 625·25-s + 7.75e3·26-s − 485·27-s + 1.56e3·28-s − 4.98e3·29-s − 200·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.0641·3-s + 4-s + 0.447·5-s − 0.0907·6-s + 0.377·7-s − 0.995·9-s − 0.632·10-s − 1.12·11-s + 0.0641·12-s − 1.59·13-s − 0.534·14-s + 0.0286·15-s − 16-s + 1.37·17-s + 1.40·18-s − 0.985·19-s + 0.447·20-s + 0.0242·21-s + 1.59·22-s − 0.651·23-s + 1/5·25-s + 2.24·26-s − 0.128·27-s + 0.377·28-s − 1.10·29-s − 0.0405·30-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: yes
Arithmetic: yes
Character: $\chi_{35} (1, \cdot )$
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 \( 1 - p^{2} T \)
7 \( 1 - p^{2} T \)
good2 \( 1 + p^{3} T + p^{5} T^{2} \)
3 \( 1 - T + p^{5} T^{2} \)
11 \( 1 + 453 T + p^{5} T^{2} \)
13 \( 1 + 969 T + p^{5} T^{2} \)
17 \( 1 - 1637 T + p^{5} T^{2} \)
19 \( 1 + 1550 T + p^{5} T^{2} \)
23 \( 1 + 1654 T + p^{5} T^{2} \)
29 \( 1 + 4985 T + p^{5} T^{2} \)
31 \( 1 - 1192 T + p^{5} T^{2} \)
37 \( 1 + 11018 T + p^{5} T^{2} \)
41 \( 1 + 1728 T + p^{5} T^{2} \)
43 \( 1 + 10814 T + p^{5} T^{2} \)
47 \( 1 - 26237 T + p^{5} T^{2} \)
53 \( 1 - 25936 T + p^{5} T^{2} \)
59 \( 1 + 4580 T + p^{5} T^{2} \)
61 \( 1 + 12488 T + p^{5} T^{2} \)
67 \( 1 + 15848 T + p^{5} T^{2} \)
71 \( 1 - 51792 T + p^{5} T^{2} \)
73 \( 1 - 4846 T + p^{5} T^{2} \)
79 \( 1 - 62765 T + p^{5} T^{2} \)
83 \( 1 + 23644 T + p^{5} T^{2} \)
89 \( 1 + 147300 T + p^{5} T^{2} \)
97 \( 1 + 8343 T + p^{5} T^{2} \)
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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

−15.07668629658918242530492204533, −13.87874828629611463272316414379, −12.19534746224870529542228070542, −10.68326794718356656115578889327, −9.799569404919037975018043906064, −8.457587789775511144374168805823, −7.45126682656621643384011516089, −5.36078758673093514145772144574, −2.28257279434830655435142918935, 0, 2.28257279434830655435142918935, 5.36078758673093514145772144574, 7.45126682656621643384011516089, 8.457587789775511144374168805823, 9.799569404919037975018043906064, 10.68326794718356656115578889327, 12.19534746224870529542228070542, 13.87874828629611463272316414379, 15.07668629658918242530492204533

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