Properties

Label 2-117670-1.1-c1-0-11
Degree $2$
Conductor $117670$
Sign $-1$
Analytic cond. $939.599$
Root an. cond. $30.6528$
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 + 4-s − 5-s + 7-s + 8-s − 3·9-s − 10-s − 4·11-s + 6·13-s + 14-s + 16-s − 2·17-s − 3·18-s − 20-s − 4·22-s + 25-s + 6·26-s + 28-s − 6·29-s + 8·31-s + 32-s − 2·34-s − 35-s − 3·36-s − 10·37-s − 40-s + 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 9-s − 0.316·10-s − 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.223·20-s − 0.852·22-s + 1/5·25-s + 1.17·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.169·35-s − 1/2·36-s − 1.64·37-s − 0.158·40-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117670\)    =    \(2 \cdot 5 \cdot 7 \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(939.599\)
Root analytic conductor: \(30.6528\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 117670,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
41 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
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

−13.85794115542236, −13.45665288389724, −12.90899083024432, −12.41067883787437, −11.92039794704288, −11.26462214063963, −11.08495594034025, −10.70536900059630, −10.14316254108667, −9.335023571857829, −8.743700751832752, −8.349533429581089, −7.911024787044637, −7.489902343542224, −6.612807662961658, −6.294091909268294, −5.725107415735820, −5.122327915311194, −4.837018755556024, −4.015150524644884, −3.487370214172761, −3.084598707452445, −2.350156223280730, −1.775738475158031, −0.8631895797776376, 0, 0.8631895797776376, 1.775738475158031, 2.350156223280730, 3.084598707452445, 3.487370214172761, 4.015150524644884, 4.837018755556024, 5.122327915311194, 5.725107415735820, 6.294091909268294, 6.612807662961658, 7.489902343542224, 7.911024787044637, 8.349533429581089, 8.743700751832752, 9.335023571857829, 10.14316254108667, 10.70536900059630, 11.08495594034025, 11.26462214063963, 11.92039794704288, 12.41067883787437, 12.90899083024432, 13.45665288389724, 13.85794115542236

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