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Properties

Label 2-124950-1.1-c1-0-104
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 − 6·13-s + 16-s + 17-s − 18-s + 8·23-s + 24-s + 6·26-s − 27-s − 6·29-s + 8·31-s − 32-s − 34-s + 36-s − 10·37-s + 6·39-s + 6·41-s − 12·43-s − 8·46-s − 48-s − 51-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 − 1.66·13-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.66·23-s + 0.204·24-s + 1.17·26-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s − 1.64·37-s + 0.960·39-s + 0.937·41-s − 1.82·43-s − 1.17·46-s − 0.144·48-s − 0.140·51-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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.57444211134153, −13.28991871287037, −12.65728395587707, −12.14016352352157, −11.82645165809266, −11.40870500811424, −10.75536375558269, −10.27320277638591, −10.01932683763742, −9.410388186287193, −8.921184823980871, −8.491997626525850, −7.666450170502814, −7.398550584303151, −6.891309475291304, −6.488313526815113, −5.710942915963535, −5.183860040998737, −4.854270140480011, −4.157019755309463, −3.308921027747656, −2.817193913636476, −2.135040614126962, −1.468873123961906, −0.7016623732803197, 0, 0.7016623732803197, 1.468873123961906, 2.135040614126962, 2.817193913636476, 3.308921027747656, 4.157019755309463, 4.854270140480011, 5.183860040998737, 5.710942915963535, 6.488313526815113, 6.891309475291304, 7.398550584303151, 7.666450170502814, 8.491997626525850, 8.921184823980871, 9.410388186287193, 10.01932683763742, 10.27320277638591, 10.75536375558269, 11.40870500811424, 11.82645165809266, 12.14016352352157, 12.65728395587707, 13.28991871287037, 13.57444211134153

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