Properties

Label 2-90354-1.1-c1-0-14
Degree $2$
Conductor $90354$
Sign $-1$
Analytic cond. $721.480$
Root an. cond. $26.8603$
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  + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s + 9-s − 11-s + 12-s − 4·13-s − 2·14-s + 16-s − 2·17-s + 18-s + 8·19-s − 2·21-s − 22-s + 2·23-s + 24-s − 5·25-s − 4·26-s + 27-s − 2·28-s + 2·29-s + 8·31-s + 32-s − 33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.83·19-s − 0.436·21-s − 0.213·22-s + 0.417·23-s + 0.204·24-s − 25-s − 0.784·26-s + 0.192·27-s − 0.377·28-s + 0.371·29-s + 1.43·31-s + 0.176·32-s − 0.174·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90354\)    =    \(2 \cdot 3 \cdot 11 \cdot 37^{2}\)
Sign: $-1$
Analytic conductor: \(721.480\)
Root analytic conductor: \(26.8603\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 90354,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 - T \)
11 \( 1 + T \)
37 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.93138168325349, −13.63230762727429, −13.17697751718090, −12.88040769143233, −12.02205237998431, −11.81120675149799, −11.46291111808346, −10.47947648917406, −9.976964698311359, −9.868805659241990, −9.166419655956386, −8.621450651263937, −7.857510439620235, −7.541832635596587, −7.001600989663615, −6.472747804864588, −5.908541276166104, −5.211562354616293, −4.779792152420853, −4.241774852538048, −3.389726258271705, −3.073024091835415, −2.570350460218261, −1.859273133984862, −1.006414337794552, 0, 1.006414337794552, 1.859273133984862, 2.570350460218261, 3.073024091835415, 3.389726258271705, 4.241774852538048, 4.779792152420853, 5.211562354616293, 5.908541276166104, 6.472747804864588, 7.001600989663615, 7.541832635596587, 7.857510439620235, 8.621450651263937, 9.166419655956386, 9.868805659241990, 9.976964698311359, 10.47947648917406, 11.46291111808346, 11.81120675149799, 12.02205237998431, 12.88040769143233, 13.17697751718090, 13.63230762727429, 13.93138168325349

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