Properties

Label 2-35-35.34-c8-0-22
Degree $2$
Conductor $35$
Sign $1$
Analytic cond. $14.2582$
Root an. cond. $3.77600$
Motivic weight $8$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 127·3-s + 256·4-s + 625·5-s + 2.40e3·7-s + 9.56e3·9-s − 2.39e4·11-s + 3.25e4·12-s − 5.65e4·13-s + 7.93e4·15-s + 6.55e4·16-s − 9.78e4·17-s + 1.60e5·20-s + 3.04e5·21-s + 3.90e5·25-s + 3.81e5·27-s + 6.14e5·28-s − 8.51e4·29-s − 3.04e6·33-s + 1.50e6·35-s + 2.44e6·36-s − 7.18e6·39-s − 6.13e6·44-s + 5.98e6·45-s + 2.19e6·47-s + 8.32e6·48-s + 5.76e6·49-s − 1.24e7·51-s + ⋯
L(s)  = 1  + 1.56·3-s + 4-s + 5-s + 7-s + 1.45·9-s − 1.63·11-s + 1.56·12-s − 1.98·13-s + 1.56·15-s + 16-s − 1.17·17-s + 20-s + 1.56·21-s + 25-s + 0.718·27-s + 28-s − 0.120·29-s − 2.56·33-s + 35-s + 1.45·36-s − 3.10·39-s − 1.63·44-s + 1.45·45-s + 0.449·47-s + 1.56·48-s + 49-s − 1.83·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $1$
Analytic conductor: \(14.2582\)
Root analytic conductor: \(3.77600\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: $\chi_{35} (34, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(4.096508646\)
\(L(\frac12)\) \(\approx\) \(4.096508646\)
\(L(5)\) 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 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
3 \( 1 - 127 T + p^{8} T^{2} \)
11 \( 1 + 23953 T + p^{8} T^{2} \)
13 \( 1 + 56593 T + p^{8} T^{2} \)
17 \( 1 + 97873 T + p^{8} T^{2} \)
19 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
23 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
29 \( 1 + 85153 T + p^{8} T^{2} \)
31 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
37 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
41 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
43 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
47 \( 1 - 2191487 T + p^{8} T^{2} \)
53 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
59 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
61 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
67 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
71 \( 1 - 50742722 T + p^{8} T^{2} \)
73 \( 1 - 33491522 T + p^{8} T^{2} \)
79 \( 1 - 70135727 T + p^{8} T^{2} \)
83 \( 1 + 54186718 T + p^{8} T^{2} \)
89 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
97 \( 1 + 165613873 T + p^{8} T^{2} \)
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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

−14.78321163555055307432545533322, −13.78449507576360099158481522691, −12.60132151440288976407849314495, −10.77329670271126327074722072734, −9.666986337406237383759137489657, −8.163674853165406947662859530180, −7.20744615196774924019256189907, −5.08440593572480679415109835650, −2.53840175613997127757002273062, −2.14662507106439729136376699922, 2.14662507106439729136376699922, 2.53840175613997127757002273062, 5.08440593572480679415109835650, 7.20744615196774924019256189907, 8.163674853165406947662859530180, 9.666986337406237383759137489657, 10.77329670271126327074722072734, 12.60132151440288976407849314495, 13.78449507576360099158481522691, 14.78321163555055307432545533322

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