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Properties

Label 2-114950-1.1-c1-0-76
Degree $2$
Conductor $114950$
Sign $-1$
Analytic cond. $917.880$
Root an. cond. $30.2965$
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 + 3·3-s + 4-s + 3·6-s − 5·7-s + 8-s + 6·9-s + 3·12-s − 13-s − 5·14-s + 16-s − 3·17-s + 6·18-s − 19-s − 15·21-s − 7·23-s + 3·24-s − 26-s + 9·27-s − 5·28-s + 3·29-s − 2·31-s + 32-s − 3·34-s + 6·36-s + 2·37-s − 38-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.22·6-s − 1.88·7-s + 0.353·8-s + 2·9-s + 0.866·12-s − 0.277·13-s − 1.33·14-s + 1/4·16-s − 0.727·17-s + 1.41·18-s − 0.229·19-s − 3.27·21-s − 1.45·23-s + 0.612·24-s − 0.196·26-s + 1.73·27-s − 0.944·28-s + 0.557·29-s − 0.359·31-s + 0.176·32-s − 0.514·34-s + 36-s + 0.328·37-s − 0.162·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114950\)    =    \(2 \cdot 5^{2} \cdot 11^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(917.880\)
Root analytic conductor: \(30.2965\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 114950,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
11 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - p T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
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

−13.80612498969115, −13.43254799117089, −13.01179795310394, −12.59735325658077, −12.26438868756103, −11.61718877914303, −10.76074716705862, −10.32680345047273, −9.779340285656072, −9.504820293198602, −8.953310279089933, −8.510452048341071, −7.840635941261759, −7.406820087879913, −6.835632028234274, −6.421392034243830, −5.916579320980820, −5.200104546237290, −4.236947169642165, −3.951420816006257, −3.619191656526073, −2.795519389353875, −2.520315691680941, −2.110811842637260, −1.036874822313901, 0, 1.036874822313901, 2.110811842637260, 2.520315691680941, 2.795519389353875, 3.619191656526073, 3.951420816006257, 4.236947169642165, 5.200104546237290, 5.916579320980820, 6.421392034243830, 6.835632028234274, 7.406820087879913, 7.840635941261759, 8.510452048341071, 8.953310279089933, 9.504820293198602, 9.779340285656072, 10.32680345047273, 10.76074716705862, 11.61718877914303, 12.26438868756103, 12.59735325658077, 13.01179795310394, 13.43254799117089, 13.80612498969115

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