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Properties

Label 2-124950-1.1-c1-0-148
Degree $2$
Conductor $124950$
Sign $-1$
Analytic cond. $997.730$
Root an. cond. $31.5868$
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 + 12-s − 4·13-s + 16-s − 17-s − 18-s − 2·19-s + 9·23-s − 24-s + 4·26-s + 27-s − 2·31-s − 32-s + 34-s + 36-s + 7·37-s + 2·38-s − 4·39-s − 3·41-s + 4·43-s − 9·46-s − 6·47-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 + 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.458·19-s + 1.87·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 0.359·31-s − 0.176·32-s + 0.171·34-s + 1/6·36-s + 1.15·37-s + 0.324·38-s − 0.640·39-s − 0.468·41-s + 0.609·43-s − 1.32·46-s − 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(997.730\)
Root analytic conductor: \(31.5868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 124950,\ (\ :1/2),\ -1)\)

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 \( 1 \)
17 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.67304749589805, −13.26857640234412, −12.81015032758679, −12.32976490206262, −11.82286617117863, −11.20653535663245, −10.82010482586717, −10.32347905701134, −9.723932498853554, −9.380551471210411, −8.894555553524985, −8.490724452643030, −7.842220218813161, −7.425798098719190, −6.967701413294395, −6.533802136565669, −5.796645724446680, −5.180313671119490, −4.601354191460919, −4.101681612272763, −3.201778555813757, −2.820615434906059, −2.263348251723115, −1.577392782220141, −0.8542626131156765, 0, 0.8542626131156765, 1.577392782220141, 2.263348251723115, 2.820615434906059, 3.201778555813757, 4.101681612272763, 4.601354191460919, 5.180313671119490, 5.796645724446680, 6.533802136565669, 6.967701413294395, 7.425798098719190, 7.842220218813161, 8.490724452643030, 8.894555553524985, 9.380551471210411, 9.723932498853554, 10.32347905701134, 10.82010482586717, 11.20653535663245, 11.82286617117863, 12.32976490206262, 12.81015032758679, 13.26857640234412, 13.67304749589805

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