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Properties

Label 2-303450-1.1-c1-0-122
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 − 3·11-s − 12-s − 14-s + 16-s + 18-s + 21-s − 3·22-s − 23-s − 24-s − 27-s − 28-s + 2·29-s + 10·31-s + 32-s + 3·33-s + 36-s − 37-s + 10·41-s + 42-s − 9·43-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 − 0.904·11-s − 0.288·12-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.218·21-s − 0.639·22-s − 0.208·23-s − 0.204·24-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + 1.79·31-s + 0.176·32-s + 0.522·33-s + 1/6·36-s − 0.164·37-s + 1.56·41-s + 0.154·42-s − 1.37·43-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: \(1\)
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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−12.84196360570622, −12.48921089762234, −12.05714087694658, −11.59966590139125, −11.16314209553966, −10.70245303636981, −10.12247330479780, −9.987610252101241, −9.378167899283825, −8.635064769392365, −8.204788005977788, −7.742814387741430, −7.191771586306573, −6.669094421739069, −6.319427897365754, −5.779729173886739, −5.335241363462508, −4.852179441711187, −4.394056815039033, −3.847466317421715, −3.259263122364188, −2.579755405526555, −2.354751994528841, −1.389426087253395, −0.7927266312714080, 0, 0.7927266312714080, 1.389426087253395, 2.354751994528841, 2.579755405526555, 3.259263122364188, 3.847466317421715, 4.394056815039033, 4.852179441711187, 5.335241363462508, 5.779729173886739, 6.319427897365754, 6.669094421739069, 7.191771586306573, 7.742814387741430, 8.204788005977788, 8.635064769392365, 9.378167899283825, 9.987610252101241, 10.12247330479780, 10.70245303636981, 11.16314209553966, 11.59966590139125, 12.05714087694658, 12.48921089762234, 12.84196360570622

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