Properties

Label 2-116886-1.1-c1-0-27
Degree $2$
Conductor $116886$
Sign $-1$
Analytic cond. $933.339$
Root an. cond. $30.5506$
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 − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 12-s − 3·13-s − 14-s − 15-s + 16-s − 18-s + 6·19-s − 20-s + 21-s + 23-s − 24-s − 4·25-s + 3·26-s + 27-s + 28-s + 9·29-s + 30-s − 4·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.832·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s + 0.208·23-s − 0.204·24-s − 4/5·25-s + 0.588·26-s + 0.192·27-s + 0.188·28-s + 1.67·29-s + 0.182·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116886\)    =    \(2 \cdot 3 \cdot 7 \cdot 11^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(933.339\)
Root analytic conductor: \(30.5506\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{116886} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 116886,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 \)
23 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−13.86967070944974, −13.50532146074777, −12.70848181703579, −12.29959587483029, −11.87356054112062, −11.40564703457441, −10.89186678161367, −10.34428265595924, −9.732391330013985, −9.533032975819494, −8.916134387761322, −8.389758458815537, −7.783649567563127, −7.675494258674824, −7.027997805089503, −6.581739114594878, −5.741022612025718, −5.292171967697605, −4.525423801391443, −4.170264801048318, −3.175981117399960, −2.995249410501804, −2.211464506858851, −1.537286878463301, −0.8928179916891668, 0, 0.8928179916891668, 1.537286878463301, 2.211464506858851, 2.995249410501804, 3.175981117399960, 4.170264801048318, 4.525423801391443, 5.292171967697605, 5.741022612025718, 6.581739114594878, 7.027997805089503, 7.675494258674824, 7.783649567563127, 8.389758458815537, 8.916134387761322, 9.533032975819494, 9.732391330013985, 10.34428265595924, 10.89186678161367, 11.40564703457441, 11.87356054112062, 12.29959587483029, 12.70848181703579, 13.50532146074777, 13.86967070944974

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