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Properties

Label 2-139650-1.1-c1-0-112
Degree $2$
Conductor $139650$
Sign $-1$
Analytic cond. $1115.11$
Root an. cond. $33.3932$
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 − 8-s + 9-s − 4·11-s + 12-s − 6·13-s + 16-s − 2·17-s − 18-s − 19-s + 4·22-s − 24-s + 6·26-s + 27-s + 2·29-s + 8·31-s − 32-s − 4·33-s + 2·34-s + 36-s + 2·37-s + 38-s − 6·39-s + 10·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 1.66·13-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.229·19-s + 0.852·22-s − 0.204·24-s + 1.17·26-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.176·32-s − 0.696·33-s + 0.342·34-s + 1/6·36-s + 0.328·37-s + 0.162·38-s − 0.960·39-s + 1.56·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(139650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(1115.11\)
Root analytic conductor: \(33.3932\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.72734316488963, −13.01878370461553, −12.72637017210644, −12.26679438277490, −11.73335440551167, −11.11886367847209, −10.64913808526851, −10.12691731533827, −9.868234973691529, −9.250316106167568, −8.892043208135783, −8.209847603896171, −7.778773128114980, −7.517611882946165, −6.950937528940235, −6.351441667793746, −5.744458801204355, −5.145146602482874, −4.409045225211697, −4.281087038010105, −2.984610381447905, −2.778503896566713, −2.362033772165514, −1.624976288062681, −0.7174258352345235, 0, 0.7174258352345235, 1.624976288062681, 2.362033772165514, 2.778503896566713, 2.984610381447905, 4.281087038010105, 4.409045225211697, 5.145146602482874, 5.744458801204355, 6.351441667793746, 6.950937528940235, 7.517611882946165, 7.778773128114980, 8.209847603896171, 8.892043208135783, 9.250316106167568, 9.868234973691529, 10.12691731533827, 10.64913808526851, 11.11886367847209, 11.73335440551167, 12.26679438277490, 12.72637017210644, 13.01878370461553, 13.72734316488963

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