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Properties

Label 2-303450-1.1-c1-0-62
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 + 5·13-s + 14-s + 16-s − 18-s + 2·19-s + 21-s − 4·22-s − 3·23-s + 24-s − 5·26-s − 27-s − 28-s + 8·29-s + 6·31-s − 32-s − 4·33-s + 36-s + 37-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 + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.458·19-s + 0.218·21-s − 0.852·22-s − 0.625·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s − 0.188·28-s + 1.48·29-s + 1.07·31-s − 0.176·32-s − 0.696·33-s + 1/6·36-s + 0.164·37-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

\(L(1)\) \(\approx\) \(2.199029164\)
\(L(\frac12)\) \(\approx\) \(2.199029164\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−12.50487983397450, −12.03528230772609, −11.75350553436348, −11.31998030541539, −10.81481033940600, −10.32298375026147, −10.05478315553848, −9.370764214645245, −9.051369984613982, −8.634039074716019, −8.074891999306370, −7.599976216244607, −7.039190904476509, −6.459440907629924, −6.152908063276028, −5.967653352318863, −5.151006590442446, −4.454027452905297, −4.053665426600552, −3.499639524681135, −2.902411775371101, −2.268589858801701, −1.419056196189427, −1.052878118729228, −0.5562752828040814, 0.5562752828040814, 1.052878118729228, 1.419056196189427, 2.268589858801701, 2.902411775371101, 3.499639524681135, 4.053665426600552, 4.454027452905297, 5.151006590442446, 5.967653352318863, 6.152908063276028, 6.459440907629924, 7.039190904476509, 7.599976216244607, 8.074891999306370, 8.634039074716019, 9.051369984613982, 9.370764214645245, 10.05478315553848, 10.32298375026147, 10.81481033940600, 11.31998030541539, 11.75350553436348, 12.03528230772609, 12.50487983397450

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