Properties

Label 2-148830-1.1-c1-0-31
Degree $2$
Conductor $148830$
Sign $1$
Analytic cond. $1188.41$
Root an. cond. $34.4733$
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 − 8-s + 9-s − 10-s + 12-s + 2·13-s + 15-s + 16-s + 6·17-s − 18-s + 4·19-s + 20-s − 8·23-s − 24-s + 25-s − 2·26-s + 27-s − 6·29-s − 30-s − 32-s − 6·34-s + 36-s + 6·37-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 + 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 1.11·29-s − 0.182·30-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.986·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(148830\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 41\)
Sign: $1$
Analytic conductor: \(1188.41\)
Root analytic conductor: \(34.4733\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 148830,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.206481221\)
\(L(\frac12)\) \(\approx\) \(3.206481221\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 \)
41 \( 1 + T \)
good7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.46295551642510, −12.77261654101252, −12.47978855909099, −11.78979580877350, −11.47165503369364, −10.86407855643424, −10.27264379342161, −9.851703650794293, −9.593298218516343, −9.068750274045802, −8.518905560540409, −7.943695546703373, −7.635417871126609, −7.271728723995118, −6.377280094055862, −6.031320019417452, −5.553176804412503, −4.940901064405211, −4.105767090132002, −3.516761442885660, −3.194768896037547, −2.303217831987339, −1.923380118744959, −1.181270146221629, −0.6168856396743082, 0.6168856396743082, 1.181270146221629, 1.923380118744959, 2.303217831987339, 3.194768896037547, 3.516761442885660, 4.105767090132002, 4.940901064405211, 5.553176804412503, 6.031320019417452, 6.377280094055862, 7.271728723995118, 7.635417871126609, 7.943695546703373, 8.518905560540409, 9.068750274045802, 9.593298218516343, 9.851703650794293, 10.27264379342161, 10.86407855643424, 11.47165503369364, 11.78979580877350, 12.47978855909099, 12.77261654101252, 13.46295551642510

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