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Properties

Label 2-323400-1.1-c1-0-117
Degree $2$
Conductor $323400$
Sign $-1$
Analytic cond. $2582.36$
Root an. cond. $50.8169$
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  − 3-s + 9-s − 11-s − 17-s − 19-s − 5·23-s − 27-s − 6·29-s + 6·31-s + 33-s − 9·37-s + 5·41-s − 4·43-s + 7·47-s + 51-s + 4·53-s + 57-s − 9·59-s + 12·61-s − 4·67-s + 5·69-s + 15·71-s − 4·73-s − 11·79-s + 81-s − 6·83-s + 6·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.301·11-s − 0.242·17-s − 0.229·19-s − 1.04·23-s − 0.192·27-s − 1.11·29-s + 1.07·31-s + 0.174·33-s − 1.47·37-s + 0.780·41-s − 0.609·43-s + 1.02·47-s + 0.140·51-s + 0.549·53-s + 0.132·57-s − 1.17·59-s + 1.53·61-s − 0.488·67-s + 0.601·69-s + 1.78·71-s − 0.468·73-s − 1.23·79-s + 1/9·81-s − 0.658·83-s + 0.643·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(323400\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(2582.36\)
Root analytic conductor: \(50.8169\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 323400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−12.80434617128932, −12.33568139745278, −11.98150183872049, −11.40472216484890, −11.13565119169444, −10.56163347939318, −10.08870747799267, −9.858250826932271, −9.218378649729252, −8.551861253363918, −8.431316433289415, −7.594399713781246, −7.336767891421081, −6.795318387989386, −6.187044670275991, −5.881302629530333, −5.322673534160138, −4.871677805419593, −4.255895670517766, −3.858021680948743, −3.268212943860785, −2.517527551012389, −2.042551058185971, −1.433429055376299, −0.6355609144728980, 0, 0.6355609144728980, 1.433429055376299, 2.042551058185971, 2.517527551012389, 3.268212943860785, 3.858021680948743, 4.255895670517766, 4.871677805419593, 5.322673534160138, 5.881302629530333, 6.187044670275991, 6.795318387989386, 7.336767891421081, 7.594399713781246, 8.431316433289415, 8.551861253363918, 9.218378649729252, 9.858250826932271, 10.08870747799267, 10.56163347939318, 11.13565119169444, 11.40472216484890, 11.98150183872049, 12.33568139745278, 12.80434617128932

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