Properties

Label 2-33635-1.1-c1-0-5
Degree $2$
Conductor $33635$
Sign $-1$
Analytic cond. $268.576$
Root an. cond. $16.3883$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 5-s + 7-s − 2·9-s + 3·11-s + 2·12-s − 5·13-s + 15-s + 4·16-s − 3·17-s + 2·19-s + 2·20-s − 21-s + 6·23-s + 25-s + 5·27-s − 2·28-s − 3·29-s − 3·33-s − 35-s + 4·36-s − 2·37-s + 5·39-s − 12·41-s + 10·43-s − 6·44-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s − 0.727·17-s + 0.458·19-s + 0.447·20-s − 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.377·28-s − 0.557·29-s − 0.522·33-s − 0.169·35-s + 2/3·36-s − 0.328·37-s + 0.800·39-s − 1.87·41-s + 1.52·43-s − 0.904·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33635\)    =    \(5 \cdot 7 \cdot 31^{2}\)
Sign: $-1$
Analytic conductor: \(268.576\)
Root analytic conductor: \(16.3883\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{33635} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 33635,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 - T \)
31 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−15.20170898618481, −14.57510144955188, −14.32017023845852, −13.76668016482181, −13.14590541481299, −12.45821777778913, −12.12809012613164, −11.63565191426304, −11.04405876414593, −10.59422274501715, −9.848305938485446, −9.202646778303202, −8.926174759869758, −8.365302518926808, −7.533875624733144, −7.213596363935943, −6.412135705243611, −5.756793863838243, −5.047407664470216, −4.785033194685816, −4.155849758818111, −3.360938239625395, −2.743499524316247, −1.670038737944617, −0.7444006095250255, 0, 0.7444006095250255, 1.670038737944617, 2.743499524316247, 3.360938239625395, 4.155849758818111, 4.785033194685816, 5.047407664470216, 5.756793863838243, 6.412135705243611, 7.213596363935943, 7.533875624733144, 8.365302518926808, 8.926174759869758, 9.202646778303202, 9.848305938485446, 10.59422274501715, 11.04405876414593, 11.63565191426304, 12.12809012613164, 12.45821777778913, 13.14590541481299, 13.76668016482181, 14.32017023845852, 14.57510144955188, 15.20170898618481

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