Properties

Label 2-40310-1.1-c1-0-14
Degree $2$
Conductor $40310$
Sign $-1$
Analytic cond. $321.876$
Root an. cond. $17.9409$
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 − 5-s − 6-s − 7-s − 8-s − 2·9-s + 10-s + 12-s + 5·13-s + 14-s − 15-s + 16-s + 3·17-s + 2·18-s − 19-s − 20-s − 21-s + 6·23-s − 24-s + 25-s − 5·26-s − 5·27-s − 28-s + 29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.229·19-s − 0.223·20-s − 0.218·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s − 0.962·27-s − 0.188·28-s + 0.185·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40310\)    =    \(2 \cdot 5 \cdot 29 \cdot 139\)
Sign: $-1$
Analytic conductor: \(321.876\)
Root analytic conductor: \(17.9409\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 40310,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 + T \)
29 \( 1 - T \)
139 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−15.08245724237234, −14.52540842514501, −14.19043269449803, −13.37902202048384, −13.01527033659800, −12.49240145489634, −11.61707891107850, −11.39854978983076, −10.86069684407156, −10.25861859658949, −9.678325819940330, −9.009348165033821, −8.676551353971095, −8.289879579898745, −7.622364986279671, −7.126710334684297, −6.458055610372899, −5.848663080147633, −5.358063355799316, −4.373221757511192, −3.639885980585048, −3.164961442584680, −2.674270552264366, −1.663532071015202, −0.9773325371848149, 0, 0.9773325371848149, 1.663532071015202, 2.674270552264366, 3.164961442584680, 3.639885980585048, 4.373221757511192, 5.358063355799316, 5.848663080147633, 6.458055610372899, 7.126710334684297, 7.622364986279671, 8.289879579898745, 8.676551353971095, 9.009348165033821, 9.678325819940330, 10.25861859658949, 10.86069684407156, 11.39854978983076, 11.61707891107850, 12.49240145489634, 13.01527033659800, 13.37902202048384, 14.19043269449803, 14.52540842514501, 15.08245724237234

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