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Properties

Label 2-139650-1.1-c1-0-120
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 − 5·11-s + 12-s − 13-s + 16-s − 3·17-s − 18-s + 19-s + 5·22-s + 23-s − 24-s + 26-s + 27-s − 29-s + 2·31-s − 32-s − 5·33-s + 3·34-s + 36-s − 4·37-s − 38-s − 39-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 − 1.50·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.229·19-s + 1.06·22-s + 0.208·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.185·29-s + 0.359·31-s − 0.176·32-s − 0.870·33-s + 0.514·34-s + 1/6·36-s − 0.657·37-s − 0.162·38-s − 0.160·39-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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−13.83379728883225, −13.04806432964559, −12.68121581175234, −12.40796888910971, −11.57149823021860, −11.11970749979733, −10.67623651601200, −10.30060313360159, −9.695597365493514, −9.293664535401665, −8.845614840287856, −8.241992263290151, −7.809906531025224, −7.519795944249558, −6.909688806287205, −6.356725837297883, −5.733778729371181, −5.145525598911368, −4.615833548908230, −4.010756104443371, −3.118542362746295, −2.836104624507484, −2.198631675289012, −1.669309301691880, −0.7369344418774093, 0, 0.7369344418774093, 1.669309301691880, 2.198631675289012, 2.836104624507484, 3.118542362746295, 4.010756104443371, 4.615833548908230, 5.145525598911368, 5.733778729371181, 6.356725837297883, 6.909688806287205, 7.519795944249558, 7.809906531025224, 8.241992263290151, 8.845614840287856, 9.293664535401665, 9.695597365493514, 10.30060313360159, 10.67623651601200, 11.11970749979733, 11.57149823021860, 12.40796888910971, 12.68121581175234, 13.04806432964559, 13.83379728883225

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