Properties

Label 2-35-35.34-c16-0-25
Degree $2$
Conductor $35$
Sign $1$
Analytic cond. $56.8135$
Root an. cond. $7.53747$
Motivic weight $16$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.00e3·3-s + 6.55e4·4-s − 3.90e5·5-s − 5.76e6·7-s − 3.40e7·9-s + 1.45e8·11-s − 1.97e8·12-s − 1.57e9·13-s + 1.17e9·15-s + 4.29e9·16-s + 4.37e9·17-s − 2.56e10·20-s + 1.73e10·21-s + 1.52e11·25-s + 2.31e11·27-s − 3.77e11·28-s − 9.93e11·29-s − 4.36e11·33-s + 2.25e12·35-s − 2.22e12·36-s + 4.72e12·39-s + 9.50e12·44-s + 1.32e13·45-s + 4.28e13·47-s − 1.29e13·48-s + 3.32e13·49-s − 1.31e13·51-s + ⋯
L(s)  = 1  − 0.458·3-s + 4-s − 5-s − 7-s − 0.789·9-s + 0.676·11-s − 0.458·12-s − 1.92·13-s + 0.458·15-s + 16-s + 0.626·17-s − 20-s + 0.458·21-s + 25-s + 0.820·27-s − 28-s − 1.98·29-s − 0.310·33-s + 35-s − 0.789·36-s + 0.882·39-s + 0.676·44-s + 0.789·45-s + 1.79·47-s − 0.458·48-s + 49-s − 0.287·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $1$
Analytic conductor: \(56.8135\)
Root analytic conductor: \(7.53747\)
Motivic weight: \(16\)
Rational: yes
Arithmetic: yes
Character: $\chi_{35} (34, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :8),\ 1)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(1.010601393\)
\(L(\frac12)\) \(\approx\) \(1.010601393\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + p^{8} T \)
7 \( 1 + p^{8} T \)
good2 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
3 \( 1 + 3007 T + p^{16} T^{2} \)
11 \( 1 - 145028447 T + p^{16} T^{2} \)
13 \( 1 + 1571306207 T + p^{16} T^{2} \)
17 \( 1 - 4372390753 T + p^{16} T^{2} \)
19 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
23 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
29 \( 1 + 993241792513 T + p^{16} T^{2} \)
31 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
37 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
41 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
43 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
47 \( 1 - 42819958052353 T + p^{16} T^{2} \)
53 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
59 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
61 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
67 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
71 \( 1 - 1283316773477762 T + p^{16} T^{2} \)
73 \( 1 - 491238137911678 T + p^{16} T^{2} \)
79 \( 1 - 1884802582005407 T + p^{16} T^{2} \)
83 \( 1 - 1568384056666558 T + p^{16} T^{2} \)
89 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
97 \( 1 + 11753087741306207 T + p^{16} T^{2} \)
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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

−12.42965174982264180815559038959, −11.93744208263411143943172219654, −10.83938174137742783098615716496, −9.453880401674966093786006723733, −7.66885331243854170002956989957, −6.77603038525530125503659339715, −5.45889922067686160621464393308, −3.63222337393984463315148608653, −2.48846029110469612796228745261, −0.52872920770931727746475546895, 0.52872920770931727746475546895, 2.48846029110469612796228745261, 3.63222337393984463315148608653, 5.45889922067686160621464393308, 6.77603038525530125503659339715, 7.66885331243854170002956989957, 9.453880401674966093786006723733, 10.83938174137742783098615716496, 11.93744208263411143943172219654, 12.42965174982264180815559038959

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