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Properties

Label 2-13475-1.1-c1-0-3
Degree $2$
Conductor $13475$
Sign $1$
Analytic cond. $107.598$
Root an. cond. $10.3729$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s + 9-s − 11-s + 4·12-s − 13-s + 4·16-s + 3·17-s + 19-s + 4·27-s + 4·31-s + 2·33-s − 2·36-s − 5·37-s + 2·39-s − 3·41-s − 8·43-s + 2·44-s + 12·47-s − 8·48-s − 6·51-s + 2·52-s + 9·53-s − 2·57-s + 12·59-s + 7·61-s − 8·64-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s + 1/3·9-s − 0.301·11-s + 1.15·12-s − 0.277·13-s + 16-s + 0.727·17-s + 0.229·19-s + 0.769·27-s + 0.718·31-s + 0.348·33-s − 1/3·36-s − 0.821·37-s + 0.320·39-s − 0.468·41-s − 1.21·43-s + 0.301·44-s + 1.75·47-s − 1.15·48-s − 0.840·51-s + 0.277·52-s + 1.23·53-s − 0.264·57-s + 1.56·59-s + 0.896·61-s − 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13475\)    =    \(5^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(107.598\)
Root analytic conductor: \(10.3729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 13475,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7310739925\)
\(L(\frac12)\) \(\approx\) \(0.7310739925\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−16.34131702816669, −15.79351859783334, −15.03302184613700, −14.51179629260878, −13.87203010060523, −13.39815863491962, −12.75715325825733, −12.07975271549400, −11.89309762702748, −11.09180147050006, −10.33181740561102, −10.08657081087136, −9.410992467960230, −8.481235394374071, −8.298339216727346, −7.244996881549942, −6.786461404267601, −5.742997980828879, −5.501793773998939, −4.925900660074532, −4.195525366027179, −3.458152291908658, −2.541847465748913, −1.241236408903666, −0.4673506302740867, 0.4673506302740867, 1.241236408903666, 2.541847465748913, 3.458152291908658, 4.195525366027179, 4.925900660074532, 5.501793773998939, 5.742997980828879, 6.786461404267601, 7.244996881549942, 8.298339216727346, 8.481235394374071, 9.410992467960230, 10.08657081087136, 10.33181740561102, 11.09180147050006, 11.89309762702748, 12.07975271549400, 12.75715325825733, 13.39815863491962, 13.87203010060523, 14.51179629260878, 15.03302184613700, 15.79351859783334, 16.34131702816669

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