Properties

Label 2-605-1.1-c3-0-104
Degree $2$
Conductor $605$
Sign $-1$
Analytic cond. $35.6961$
Root an. cond. $5.97462$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 2·3-s + 8·4-s − 5·5-s + 8·6-s − 6·7-s − 23·9-s − 20·10-s + 16·12-s + 38·13-s − 24·14-s − 10·15-s − 64·16-s − 26·17-s − 92·18-s − 100·19-s − 40·20-s − 12·21-s − 78·23-s + 25·25-s + 152·26-s − 100·27-s − 48·28-s + 50·29-s − 40·30-s − 108·31-s − 256·32-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.384·3-s + 4-s − 0.447·5-s + 0.544·6-s − 0.323·7-s − 0.851·9-s − 0.632·10-s + 0.384·12-s + 0.810·13-s − 0.458·14-s − 0.172·15-s − 16-s − 0.370·17-s − 1.20·18-s − 1.20·19-s − 0.447·20-s − 0.124·21-s − 0.707·23-s + 1/5·25-s + 1.14·26-s − 0.712·27-s − 0.323·28-s + 0.320·29-s − 0.243·30-s − 0.625·31-s − 1.41·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(35.6961\)
Root analytic conductor: \(5.97462\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 605,\ (\ :3/2),\ -1)\)

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 \( 1 + p T \)
11 \( 1 \)
good2 \( 1 - p^{2} T + p^{3} T^{2} \)
3 \( 1 - 2 T + p^{3} T^{2} \)
7 \( 1 + 6 T + p^{3} T^{2} \)
13 \( 1 - 38 T + p^{3} T^{2} \)
17 \( 1 + 26 T + p^{3} T^{2} \)
19 \( 1 + 100 T + p^{3} T^{2} \)
23 \( 1 + 78 T + p^{3} T^{2} \)
29 \( 1 - 50 T + p^{3} T^{2} \)
31 \( 1 + 108 T + p^{3} T^{2} \)
37 \( 1 - 266 T + p^{3} T^{2} \)
41 \( 1 + 22 T + p^{3} T^{2} \)
43 \( 1 + 442 T + p^{3} T^{2} \)
47 \( 1 + 514 T + p^{3} T^{2} \)
53 \( 1 - 2 T + p^{3} T^{2} \)
59 \( 1 - 500 T + p^{3} T^{2} \)
61 \( 1 - 518 T + p^{3} T^{2} \)
67 \( 1 - 126 T + p^{3} T^{2} \)
71 \( 1 - 412 T + p^{3} T^{2} \)
73 \( 1 - 878 T + p^{3} T^{2} \)
79 \( 1 + 600 T + p^{3} T^{2} \)
83 \( 1 + 282 T + p^{3} T^{2} \)
89 \( 1 + 150 T + p^{3} T^{2} \)
97 \( 1 - 386 T + p^{3} T^{2} \)
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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

−9.859136641615114443745606461166, −8.712171887583194838916139914552, −8.153990620441612553838563025613, −6.68607495135358755463518225545, −6.10240764845729438287117548498, −5.03793274786996369406823320201, −3.98524887702113779285395319249, −3.30783837142907774635534987041, −2.20888740482190146568794437978, 0, 2.20888740482190146568794437978, 3.30783837142907774635534987041, 3.98524887702113779285395319249, 5.03793274786996369406823320201, 6.10240764845729438287117548498, 6.68607495135358755463518225545, 8.153990620441612553838563025613, 8.712171887583194838916139914552, 9.859136641615114443745606461166

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