Properties

Label 2-309442-1.1-c1-0-8
Degree $2$
Conductor $309442$
Sign $1$
Analytic cond. $2470.90$
Root an. cond. $49.7082$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·6-s + 7-s − 8-s + 9-s − 4·11-s − 2·12-s − 14-s + 16-s − 6·17-s − 18-s − 6·19-s − 2·21-s + 4·22-s + 23-s + 2·24-s − 5·25-s + 4·27-s + 28-s − 10·29-s − 32-s + 8·33-s + 6·34-s + 36-s + 2·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.37·19-s − 0.436·21-s + 0.852·22-s + 0.208·23-s + 0.408·24-s − 25-s + 0.769·27-s + 0.188·28-s − 1.85·29-s − 0.176·32-s + 1.39·33-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 309442 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309442 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(309442\)    =    \(2 \cdot 7 \cdot 23 \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(2470.90\)
Root analytic conductor: \(49.7082\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{309442} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 309442,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 - T \)
31 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.06220115995480, −12.53564704761836, −12.05314503406002, −11.64202930718564, −11.06608963907311, −10.90705233757546, −10.51111908157108, −10.14451602402969, −9.412741726519890, −8.921884538026604, −8.614494774776864, −8.000201825848246, −7.527011800509176, −7.160695185859138, −6.407678987955483, −6.239293643385390, −5.585666477121215, −5.213087455500775, −4.712207247288884, −4.076664573597762, −3.602563969457357, −2.560032873017225, −2.266748952090286, −1.773267810966299, −0.8438522699074406, 0, 0, 0.8438522699074406, 1.773267810966299, 2.266748952090286, 2.560032873017225, 3.602563969457357, 4.076664573597762, 4.712207247288884, 5.213087455500775, 5.585666477121215, 6.239293643385390, 6.407678987955483, 7.160695185859138, 7.527011800509176, 8.000201825848246, 8.614494774776864, 8.921884538026604, 9.412741726519890, 10.14451602402969, 10.51111908157108, 10.90705233757546, 11.06608963907311, 11.64202930718564, 12.05314503406002, 12.53564704761836, 13.06220115995480

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