Properties

Label 2-11310-1.1-c1-0-7
Degree $2$
Conductor $11310$
Sign $-1$
Analytic cond. $90.3108$
Root an. cond. $9.50319$
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 − 8-s + 9-s − 10-s − 4·11-s + 12-s − 13-s + 15-s + 16-s − 4·17-s − 18-s + 20-s + 4·22-s + 2·23-s − 24-s + 25-s + 26-s + 27-s − 29-s − 30-s + 4·31-s − 32-s − 4·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.223·20-s + 0.852·22-s + 0.417·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.185·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11310\)    =    \(2 \cdot 3 \cdot 5 \cdot 13 \cdot 29\)
Sign: $-1$
Analytic conductor: \(90.3108\)
Root analytic conductor: \(9.50319\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{11310} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 11310,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 + T \)
29 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−16.83653416271372, −16.05719034549723, −15.70672819344579, −15.13524834302351, −14.57196485379686, −13.86595648907681, −13.30551587030539, −12.82698540041628, −12.24916494372010, −11.26105924473714, −10.93413322560278, −10.19346357389619, −9.710703138555735, −9.185200163744909, −8.421603568371746, −8.068388634445697, −7.303961642993433, −6.748781876927225, −6.009354380964941, −5.185241254162681, −4.557470189808198, −3.537799273504771, −2.590231009654176, −2.309418326216595, −1.213421669562855, 0, 1.213421669562855, 2.309418326216595, 2.590231009654176, 3.537799273504771, 4.557470189808198, 5.185241254162681, 6.009354380964941, 6.748781876927225, 7.303961642993433, 8.068388634445697, 8.421603568371746, 9.185200163744909, 9.710703138555735, 10.19346357389619, 10.93413322560278, 11.26105924473714, 12.24916494372010, 12.82698540041628, 13.30551587030539, 13.86595648907681, 14.57196485379686, 15.13524834302351, 15.70672819344579, 16.05719034549723, 16.83653416271372

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