Properties

Label 2-950-1.1-c5-0-97
Degree $2$
Conductor $950$
Sign $-1$
Analytic cond. $152.364$
Root an. cond. $12.3436$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 20.4·3-s + 16·4-s − 81.9·6-s + 18.9·7-s + 64·8-s + 176.·9-s + 349.·11-s − 327.·12-s − 711.·13-s + 75.6·14-s + 256·16-s − 221.·17-s + 705.·18-s + 361·19-s − 387.·21-s + 1.39e3·22-s + 662.·23-s − 1.31e3·24-s − 2.84e3·26-s + 1.36e3·27-s + 302.·28-s − 7.21e3·29-s + 5.40e3·31-s + 1.02e3·32-s − 7.15e3·33-s − 887.·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.31·3-s + 0.5·4-s − 0.929·6-s + 0.145·7-s + 0.353·8-s + 0.726·9-s + 0.870·11-s − 0.656·12-s − 1.16·13-s + 0.103·14-s + 0.250·16-s − 0.186·17-s + 0.513·18-s + 0.229·19-s − 0.191·21-s + 0.615·22-s + 0.261·23-s − 0.464·24-s − 0.825·26-s + 0.359·27-s + 0.0729·28-s − 1.59·29-s + 1.01·31-s + 0.176·32-s − 1.14·33-s − 0.131·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(152.364\)
Root analytic conductor: \(12.3436\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 950,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4T \)
5 \( 1 \)
19 \( 1 - 361T \)
good3 \( 1 + 20.4T + 243T^{2} \)
7 \( 1 - 18.9T + 1.68e4T^{2} \)
11 \( 1 - 349.T + 1.61e5T^{2} \)
13 \( 1 + 711.T + 3.71e5T^{2} \)
17 \( 1 + 221.T + 1.41e6T^{2} \)
23 \( 1 - 662.T + 6.43e6T^{2} \)
29 \( 1 + 7.21e3T + 2.05e7T^{2} \)
31 \( 1 - 5.40e3T + 2.86e7T^{2} \)
37 \( 1 + 1.97e3T + 6.93e7T^{2} \)
41 \( 1 + 3.11e3T + 1.15e8T^{2} \)
43 \( 1 + 318.T + 1.47e8T^{2} \)
47 \( 1 - 2.72e4T + 2.29e8T^{2} \)
53 \( 1 - 1.11e3T + 4.18e8T^{2} \)
59 \( 1 + 3.79e4T + 7.14e8T^{2} \)
61 \( 1 - 3.74e4T + 8.44e8T^{2} \)
67 \( 1 - 5.49e4T + 1.35e9T^{2} \)
71 \( 1 + 7.17e3T + 1.80e9T^{2} \)
73 \( 1 + 6.47e4T + 2.07e9T^{2} \)
79 \( 1 - 3.61e4T + 3.07e9T^{2} \)
83 \( 1 - 5.17e4T + 3.93e9T^{2} \)
89 \( 1 - 1.45e5T + 5.58e9T^{2} \)
97 \( 1 + 3.95e4T + 8.58e9T^{2} \)
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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

−9.030320106263371284530200187759, −7.69905186000330746030367434424, −6.89283237113185984173114475938, −6.18982834416182642654956277109, −5.32681871822788878088288475727, −4.71336109804064401555062674096, −3.72595559469754306097378865617, −2.39867039197355751635816936463, −1.16302896072277631483145537948, 0, 1.16302896072277631483145537948, 2.39867039197355751635816936463, 3.72595559469754306097378865617, 4.71336109804064401555062674096, 5.32681871822788878088288475727, 6.18982834416182642654956277109, 6.89283237113185984173114475938, 7.69905186000330746030367434424, 9.030320106263371284530200187759

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