Properties

Label 2-35-1.1-c3-0-5
Degree $2$
Conductor $35$
Sign $-1$
Analytic cond. $2.06506$
Root an. cond. $1.43703$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 8·3-s − 7·4-s − 5·5-s − 8·6-s + 7·7-s − 15·8-s + 37·9-s − 5·10-s + 12·11-s + 56·12-s − 78·13-s + 7·14-s + 40·15-s + 41·16-s − 94·17-s + 37·18-s + 40·19-s + 35·20-s − 56·21-s + 12·22-s + 32·23-s + 120·24-s + 25·25-s − 78·26-s − 80·27-s − 49·28-s + ⋯
L(s)  = 1  + 0.353·2-s − 1.53·3-s − 7/8·4-s − 0.447·5-s − 0.544·6-s + 0.377·7-s − 0.662·8-s + 1.37·9-s − 0.158·10-s + 0.328·11-s + 1.34·12-s − 1.66·13-s + 0.133·14-s + 0.688·15-s + 0.640·16-s − 1.34·17-s + 0.484·18-s + 0.482·19-s + 0.391·20-s − 0.581·21-s + 0.116·22-s + 0.290·23-s + 1.02·24-s + 1/5·25-s − 0.588·26-s − 0.570·27-s − 0.330·28-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: yes
Arithmetic: yes
Character: $\chi_{35} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 35,\ (\ :3/2),\ -1)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + p T \)
7 \( 1 - p T \)
good2 \( 1 - T + p^{3} T^{2} \)
3 \( 1 + 8 T + p^{3} T^{2} \)
11 \( 1 - 12 T + p^{3} T^{2} \)
13 \( 1 + 6 p T + p^{3} T^{2} \)
17 \( 1 + 94 T + p^{3} T^{2} \)
19 \( 1 - 40 T + p^{3} T^{2} \)
23 \( 1 - 32 T + p^{3} T^{2} \)
29 \( 1 + 50 T + p^{3} T^{2} \)
31 \( 1 + 8 p T + p^{3} T^{2} \)
37 \( 1 + 434 T + p^{3} T^{2} \)
41 \( 1 - 402 T + p^{3} T^{2} \)
43 \( 1 + 68 T + p^{3} T^{2} \)
47 \( 1 - 536 T + p^{3} T^{2} \)
53 \( 1 - 22 T + p^{3} T^{2} \)
59 \( 1 + 560 T + p^{3} T^{2} \)
61 \( 1 + 278 T + p^{3} T^{2} \)
67 \( 1 + 164 T + p^{3} T^{2} \)
71 \( 1 - 672 T + p^{3} T^{2} \)
73 \( 1 - 82 T + p^{3} T^{2} \)
79 \( 1 + 1000 T + p^{3} T^{2} \)
83 \( 1 + 448 T + p^{3} T^{2} \)
89 \( 1 + 870 T + p^{3} T^{2} \)
97 \( 1 - 1026 T + p^{3} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.50217950966324923898462215554, −14.26345646777693422232764700602, −12.70453334590557832947232791356, −11.92655234467305331455017611887, −10.74664086894993963800643426567, −9.192146462467410141721051534476, −7.18652248590761618046061851967, −5.45423844932287809114770528907, −4.44336193877597669726946649106, 0, 4.44336193877597669726946649106, 5.45423844932287809114770528907, 7.18652248590761618046061851967, 9.192146462467410141721051534476, 10.74664086894993963800643426567, 11.92655234467305331455017611887, 12.70453334590557832947232791356, 14.26345646777693422232764700602, 15.50217950966324923898462215554

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