Properties

Label 2-1075-1.1-c5-0-308
Degree $2$
Conductor $1075$
Sign $-1$
Analytic cond. $172.412$
Root an. cond. $13.1305$
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  + 8.86·2-s − 1.50·3-s + 46.5·4-s − 13.3·6-s + 124.·7-s + 129.·8-s − 240.·9-s + 590.·11-s − 69.9·12-s − 434.·13-s + 1.10e3·14-s − 343.·16-s − 1.92e3·17-s − 2.13e3·18-s + 654.·19-s − 187.·21-s + 5.23e3·22-s − 2.80e3·23-s − 194.·24-s − 3.85e3·26-s + 726.·27-s + 5.81e3·28-s − 1.45e3·29-s + 4.41e3·31-s − 7.18e3·32-s − 886.·33-s − 1.70e4·34-s + ⋯
L(s)  = 1  + 1.56·2-s − 0.0963·3-s + 1.45·4-s − 0.150·6-s + 0.962·7-s + 0.714·8-s − 0.990·9-s + 1.47·11-s − 0.140·12-s − 0.713·13-s + 1.50·14-s − 0.335·16-s − 1.61·17-s − 1.55·18-s + 0.415·19-s − 0.0926·21-s + 2.30·22-s − 1.10·23-s − 0.0688·24-s − 1.11·26-s + 0.191·27-s + 1.40·28-s − 0.321·29-s + 0.826·31-s − 1.24·32-s − 0.141·33-s − 2.53·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1075\)    =    \(5^{2} \cdot 43\)
Sign: $-1$
Analytic conductor: \(172.412\)
Root analytic conductor: \(13.1305\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1075,\ (\ :5/2),\ -1)\)

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)$
bad5 \( 1 \)
43 \( 1 + 1.84e3T \)
good2 \( 1 - 8.86T + 32T^{2} \)
3 \( 1 + 1.50T + 243T^{2} \)
7 \( 1 - 124.T + 1.68e4T^{2} \)
11 \( 1 - 590.T + 1.61e5T^{2} \)
13 \( 1 + 434.T + 3.71e5T^{2} \)
17 \( 1 + 1.92e3T + 1.41e6T^{2} \)
19 \( 1 - 654.T + 2.47e6T^{2} \)
23 \( 1 + 2.80e3T + 6.43e6T^{2} \)
29 \( 1 + 1.45e3T + 2.05e7T^{2} \)
31 \( 1 - 4.41e3T + 2.86e7T^{2} \)
37 \( 1 + 3.75e3T + 6.93e7T^{2} \)
41 \( 1 - 1.97e3T + 1.15e8T^{2} \)
47 \( 1 + 2.20e3T + 2.29e8T^{2} \)
53 \( 1 + 2.49e4T + 4.18e8T^{2} \)
59 \( 1 + 4.27e4T + 7.14e8T^{2} \)
61 \( 1 + 2.10e4T + 8.44e8T^{2} \)
67 \( 1 - 2.52e4T + 1.35e9T^{2} \)
71 \( 1 - 4.80e4T + 1.80e9T^{2} \)
73 \( 1 + 5.88e4T + 2.07e9T^{2} \)
79 \( 1 - 9.27e4T + 3.07e9T^{2} \)
83 \( 1 - 1.84e3T + 3.93e9T^{2} \)
89 \( 1 + 7.03e4T + 5.58e9T^{2} \)
97 \( 1 - 8.88e4T + 8.58e9T^{2} \)
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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

−8.732492967710803163826897321156, −7.74316987142740745775779107621, −6.59258315352168739782150098287, −6.13905400642525448999096482792, −5.05220514834546067250698491660, −4.50762848832024827923458217893, −3.64305533971625001759911613970, −2.54345866156770410617192414318, −1.66772008976538036820832934375, 0, 1.66772008976538036820832934375, 2.54345866156770410617192414318, 3.64305533971625001759911613970, 4.50762848832024827923458217893, 5.05220514834546067250698491660, 6.13905400642525448999096482792, 6.59258315352168739782150098287, 7.74316987142740745775779107621, 8.732492967710803163826897321156

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