Properties

Label 2-5415-1.1-c1-0-74
Degree $2$
Conductor $5415$
Sign $1$
Analytic cond. $43.2389$
Root an. cond. $6.57563$
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 + 5-s + 6-s − 3·8-s + 9-s + 10-s − 4·11-s − 12-s + 2·13-s + 15-s − 16-s + 2·17-s + 18-s − 20-s − 4·22-s − 3·24-s + 25-s + 2·26-s + 27-s + 2·29-s + 30-s + 5·32-s − 4·33-s + 2·34-s − 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.223·20-s − 0.852·22-s − 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.371·29-s + 0.182·30-s + 0.883·32-s − 0.696·33-s + 0.342·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5415\)    =    \(3 \cdot 5 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(43.2389\)
Root analytic conductor: \(6.57563\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5415} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5415,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.929624646\)
\(L(\frac12)\) \(\approx\) \(2.929624646\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 - T \)
19 \( 1 \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−8.179427569545746398797779230432, −7.60719004255564017514118173943, −6.56470169945492021349835720384, −5.84128873133936523436206316749, −5.21484715703970999635976715829, −4.52985320583773135666017839183, −3.66793113681489031905327880745, −2.96337185591461309202736907725, −2.19847936643903163153856477125, −0.803857922530386741108529528003, 0.803857922530386741108529528003, 2.19847936643903163153856477125, 2.96337185591461309202736907725, 3.66793113681489031905327880745, 4.52985320583773135666017839183, 5.21484715703970999635976715829, 5.84128873133936523436206316749, 6.56470169945492021349835720384, 7.60719004255564017514118173943, 8.179427569545746398797779230432

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