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Properties

Label 2-303450-1.1-c1-0-13
Degree $2$
Conductor $303450$
Sign $1$
Analytic cond. $2423.06$
Root an. cond. $49.2245$
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 + 7-s + 8-s + 9-s − 4·11-s + 12-s − 4·13-s + 14-s + 16-s + 18-s − 6·19-s + 21-s − 4·22-s + 2·23-s + 24-s − 4·26-s + 27-s + 28-s − 4·31-s + 32-s − 4·33-s + 36-s − 6·38-s − 4·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s − 1.37·19-s + 0.218·21-s − 0.852·22-s + 0.417·23-s + 0.204·24-s − 0.784·26-s + 0.192·27-s + 0.188·28-s − 0.718·31-s + 0.176·32-s − 0.696·33-s + 1/6·36-s − 0.973·38-s − 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(303450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2423.06\)
Root analytic conductor: \(49.2245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.962787451\)
\(L(\frac12)\) \(\approx\) \(1.962787451\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
17 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−12.79637752727980, −12.39871319005054, −11.81980190326485, −11.38136887446927, −10.83176639880637, −10.37469501250011, −10.08612860575391, −9.549427616692868, −8.847301167084687, −8.465472822152347, −8.018729762823153, −7.534906070932450, −7.092931386671658, −6.642624002159776, −6.087360924186823, −5.223412630485522, −5.172897835116375, −4.647396440117537, −3.989827112127846, −3.559300148722156, −2.825645549823093, −2.460641457363758, −1.998517880268992, −1.391021198259908, −0.2856325891376890, 0.2856325891376890, 1.391021198259908, 1.998517880268992, 2.460641457363758, 2.825645549823093, 3.559300148722156, 3.989827112127846, 4.647396440117537, 5.172897835116375, 5.223412630485522, 6.087360924186823, 6.642624002159776, 7.092931386671658, 7.534906070932450, 8.018729762823153, 8.465472822152347, 8.847301167084687, 9.549427616692868, 10.08612860575391, 10.37469501250011, 10.83176639880637, 11.38136887446927, 11.81980190326485, 12.39871319005054, 12.79637752727980

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