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Properties

Label 2-388815-1.1-c1-0-19
Degree $2$
Conductor $388815$
Sign $1$
Analytic cond. $3104.70$
Root an. cond. $55.7198$
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-s + 3-s − 4-s − 5-s − 6-s + 3·8-s + 9-s + 10-s − 4·11-s − 12-s + 6·13-s − 15-s − 16-s − 2·17-s − 18-s + 20-s + 4·22-s + 3·24-s + 25-s − 6·26-s + 27-s + 6·29-s + 30-s − 5·32-s − 4·33-s + 2·34-s − 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 1.66·13-s − 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.223·20-s + 0.852·22-s + 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s − 0.883·32-s − 0.696·33-s + 0.342·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(388815\)    =    \(3 \cdot 5 \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(3104.70\)
Root analytic conductor: \(55.7198\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 388815,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.517482926\)
\(L(\frac12)\) \(\approx\) \(1.517482926\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.59387005975657, −11.97483223045103, −11.36530325241693, −11.01440424146721, −10.53931049106239, −10.15499273461315, −9.788659818597682, −9.183436973136858, −8.592123446386949, −8.401674869071615, −8.218527878295463, −7.624534882493532, −7.034590366926673, −6.690465001200377, −6.000465442018271, −5.396697964466754, −4.901623853985601, −4.453280542321301, −3.862133541868539, −3.442622416558415, −2.929181891552874, −2.184378742046898, −1.643749160640780, −0.9509106957713767, −0.4167844445620186, 0.4167844445620186, 0.9509106957713767, 1.643749160640780, 2.184378742046898, 2.929181891552874, 3.442622416558415, 3.862133541868539, 4.453280542321301, 4.901623853985601, 5.396697964466754, 6.000465442018271, 6.690465001200377, 7.034590366926673, 7.624534882493532, 8.218527878295463, 8.401674869071615, 8.592123446386949, 9.183436973136858, 9.788659818597682, 10.15499273461315, 10.53931049106239, 11.01440424146721, 11.36530325241693, 11.97483223045103, 12.59387005975657

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