Properties

Label 2-308550-1.1-c1-0-34
Degree $2$
Conductor $308550$
Sign $1$
Analytic cond. $2463.78$
Root an. cond. $49.6365$
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 − 2·7-s − 8-s + 9-s + 12-s + 2·14-s + 16-s + 17-s − 18-s − 2·21-s + 4·23-s − 24-s + 27-s − 2·28-s − 2·29-s + 4·31-s − 32-s − 34-s + 36-s + 2·37-s + 2·42-s − 6·43-s − 4·46-s + 48-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.436·21-s + 0.834·23-s − 0.204·24-s + 0.192·27-s − 0.377·28-s − 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s + 0.328·37-s + 0.308·42-s − 0.914·43-s − 0.589·46-s + 0.144·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308550\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(2463.78\)
Root analytic conductor: \(49.6365\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{308550} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 308550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.529365635\)
\(L(\frac12)\) \(\approx\) \(1.529365635\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
17 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−12.66879015500768, −12.21976467981860, −11.65544821771447, −11.29875181227812, −10.71145512497535, −10.18977845581544, −9.885694140197355, −9.460989753231248, −9.008667666929270, −8.501402150591875, −8.193092110924614, −7.578278331221358, −7.136129942679209, −6.666875545265024, −6.338909535137428, −5.571503127485891, −5.211871861055249, −4.448848519524448, −3.859931381783970, −3.407267496238186, −2.703824621269193, −2.561911448870822, −1.621134002188931, −1.151436167821269, −0.3715131147786168, 0.3715131147786168, 1.151436167821269, 1.621134002188931, 2.561911448870822, 2.703824621269193, 3.407267496238186, 3.859931381783970, 4.448848519524448, 5.211871861055249, 5.571503127485891, 6.338909535137428, 6.666875545265024, 7.136129942679209, 7.578278331221358, 8.193092110924614, 8.501402150591875, 9.008667666929270, 9.460989753231248, 9.885694140197355, 10.18977845581544, 10.71145512497535, 11.29875181227812, 11.65544821771447, 12.21976467981860, 12.66879015500768

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