Properties

Label 2-303450-1.1-c1-0-95
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 $1$

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 − 11-s − 12-s − 13-s + 14-s + 16-s − 18-s + 21-s + 22-s + 4·23-s + 24-s + 26-s − 27-s − 28-s − 10·29-s − 3·31-s − 32-s + 33-s + 36-s + 2·37-s + 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 − 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.218·21-s + 0.213·22-s + 0.834·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s − 1.85·29-s − 0.538·31-s − 0.176·32-s + 0.174·33-s + 1/6·36-s + 0.328·37-s + 0.160·39-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: $\chi_{303450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
17 \( 1 \)
good11 \( 1 + T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 15 T + p T^{2} \)
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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

−12.82926526681731, −12.37437822588752, −12.03140649196275, −11.29679283501246, −10.99417314562596, −10.76390813095907, −10.19255065090136, −9.526534120346767, −9.326610081843890, −8.994969200471155, −8.214540783543288, −7.724135592702054, −7.339472718476186, −6.989438421215633, −6.318261393471559, −5.927251255680335, −5.399469389548061, −5.020630203018969, −4.219864502834417, −3.786683859559879, −3.161254103593102, −2.448442541507389, −2.085189473043908, −1.236061716404530, −0.6724641580173804, 0, 0.6724641580173804, 1.236061716404530, 2.085189473043908, 2.448442541507389, 3.161254103593102, 3.786683859559879, 4.219864502834417, 5.020630203018969, 5.399469389548061, 5.927251255680335, 6.318261393471559, 6.989438421215633, 7.339472718476186, 7.724135592702054, 8.214540783543288, 8.994969200471155, 9.326610081843890, 9.526534120346767, 10.19255065090136, 10.76390813095907, 10.99417314562596, 11.29679283501246, 12.03140649196275, 12.37437822588752, 12.82926526681731

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