Properties

Label 2-116886-1.1-c1-0-7
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 12-s − 13-s − 14-s + 16-s − 6·17-s − 18-s + 5·19-s + 21-s − 23-s − 24-s − 5·25-s + 26-s + 27-s + 28-s + 6·29-s − 31-s − 32-s + 6·34-s + 36-s + 11·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.14·19-s + 0.218·21-s − 0.208·23-s − 0.204·24-s − 25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.179·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.80·37-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: \(0\)
Selberg data: \((2,\ 116886,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.368052256\)
\(L(\frac12)\) \(\approx\) \(2.368052256\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−13.78085571010523, −13.13845956669533, −12.61906824881724, −12.04864049673349, −11.53009724270232, −11.18071223661683, −10.66138451980956, −9.964952817072973, −9.675721046945309, −9.219583289175896, −8.563379052731439, −8.329105354682435, −7.636053242864174, −7.318199737622435, −6.771997009941941, −6.141029493826330, −5.587760311871155, −4.949031357366873, −4.116431829480785, −4.011344537156384, −2.867999006660508, −2.567357350003652, −1.983796133429409, −1.188367373122896, −0.5440895098492740, 0.5440895098492740, 1.188367373122896, 1.983796133429409, 2.567357350003652, 2.867999006660508, 4.011344537156384, 4.116431829480785, 4.949031357366873, 5.587760311871155, 6.141029493826330, 6.771997009941941, 7.318199737622435, 7.636053242864174, 8.329105354682435, 8.563379052731439, 9.219583289175896, 9.675721046945309, 9.964952817072973, 10.66138451980956, 11.18071223661683, 11.53009724270232, 12.04864049673349, 12.61906824881724, 13.13845956669533, 13.78085571010523

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