Properties

Label 2-139650-1.1-c1-0-118
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 $0$

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 + 13-s + 16-s − 2·17-s + 18-s − 19-s + 4·22-s − 23-s + 24-s + 26-s + 27-s + 9·29-s + 32-s + 4·33-s − 2·34-s + 36-s − 2·37-s − 38-s + 39-s + 2·41-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 + 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.229·19-s + 0.852·22-s − 0.208·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 1.67·29-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 + 0.160·39-s + 0.312·41-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: \(0\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.417486299\)
\(L(\frac12)\) \(\approx\) \(7.417486299\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−13.62673397965624, −12.90944433070267, −12.50453586956133, −12.06174178015751, −11.67408461996282, −10.99373321911716, −10.65040727762041, −10.09918427951152, −9.439247474262894, −9.030427617775508, −8.625530199809919, −7.970104632607917, −7.548111691928862, −6.853621031022454, −6.396152870068434, −6.182783164932568, −5.298262170761159, −4.795646887033986, −4.137257094903105, −3.875017356721015, −3.257831304980631, −2.524127246440809, −2.136632391172048, −1.289942838714494, −0.7313470159378377, 0.7313470159378377, 1.289942838714494, 2.136632391172048, 2.524127246440809, 3.257831304980631, 3.875017356721015, 4.137257094903105, 4.795646887033986, 5.298262170761159, 6.182783164932568, 6.396152870068434, 6.853621031022454, 7.548111691928862, 7.970104632607917, 8.625530199809919, 9.030427617775508, 9.439247474262894, 10.09918427951152, 10.65040727762041, 10.99373321911716, 11.67408461996282, 12.06174178015751, 12.50453586956133, 12.90944433070267, 13.62673397965624

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