Properties

Label 2-30345-1.1-c1-0-13
Degree $2$
Conductor $30345$
Sign $1$
Analytic cond. $242.306$
Root an. cond. $15.5661$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 4-s + 5-s + 6-s + 7-s − 3·8-s + 9-s + 10-s + 11-s − 12-s − 13-s + 14-s + 15-s − 16-s + 18-s − 20-s + 21-s + 22-s − 3·24-s + 25-s − 26-s + 27-s − 28-s − 6·29-s + 30-s + 7·31-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 − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.235·18-s − 0.223·20-s + 0.218·21-s + 0.213·22-s − 0.612·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.182·30-s + 1.25·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30345\)    =    \(3 \cdot 5 \cdot 7 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(242.306\)
Root analytic conductor: \(15.5661\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 30345,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.901441209\)
\(L(\frac12)\) \(\approx\) \(3.901441209\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - T \)
17 \( 1 \)
good2 \( 1 - T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.95676225847255, −14.47342804813787, −14.13340085942447, −13.55920226775931, −13.08121701499926, −12.73656847612059, −12.02742192515731, −11.55383310382519, −10.89654181596087, −10.15399598952737, −9.610470288942653, −9.184969836294670, −8.733081984189003, −7.910267037624410, −7.637638260814906, −6.664033436145199, −6.102362538611007, −5.600288584091826, −4.789356314806210, −4.418042305141774, −3.788625390498179, −2.999402134956419, −2.487568338256464, −1.589616118344409, −0.6665656962130901, 0.6665656962130901, 1.589616118344409, 2.487568338256464, 2.999402134956419, 3.788625390498179, 4.418042305141774, 4.789356314806210, 5.600288584091826, 6.102362538611007, 6.664033436145199, 7.637638260814906, 7.910267037624410, 8.733081984189003, 9.184969836294670, 9.610470288942653, 10.15399598952737, 10.89654181596087, 11.55383310382519, 12.02742192515731, 12.73656847612059, 13.08121701499926, 13.55920226775931, 14.13340085942447, 14.47342804813787, 14.95676225847255

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