Properties

Label 2-8450-1.1-c1-0-133
Degree $2$
Conductor $8450$
Sign $1$
Analytic cond. $67.4735$
Root an. cond. $8.21423$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2.73·3-s + 4-s − 2.73·6-s + 3·7-s − 8-s + 4.46·9-s + 3·11-s + 2.73·12-s − 3·14-s + 16-s − 2.19·17-s − 4.46·18-s + 6.46·19-s + 8.19·21-s − 3·22-s + 2.53·23-s − 2.73·24-s + 3.99·27-s + 3·28-s − 9.46·29-s + 1.26·31-s − 32-s + 8.19·33-s + 2.19·34-s + 4.46·36-s + 11.1·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.57·3-s + 0.5·4-s − 1.11·6-s + 1.13·7-s − 0.353·8-s + 1.48·9-s + 0.904·11-s + 0.788·12-s − 0.801·14-s + 0.250·16-s − 0.532·17-s − 1.05·18-s + 1.48·19-s + 1.78·21-s − 0.639·22-s + 0.528·23-s − 0.557·24-s + 0.769·27-s + 0.566·28-s − 1.75·29-s + 0.227·31-s − 0.176·32-s + 1.42·33-s + 0.376·34-s + 0.744·36-s + 1.84·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8450\)    =    \(2 \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(67.4735\)
Root analytic conductor: \(8.21423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.684794270\)
\(L(\frac12)\) \(\approx\) \(3.684794270\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good3 \( 1 - 2.73T + 3T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
17 \( 1 + 2.19T + 17T^{2} \)
19 \( 1 - 6.46T + 19T^{2} \)
23 \( 1 - 2.53T + 23T^{2} \)
29 \( 1 + 9.46T + 29T^{2} \)
31 \( 1 - 1.26T + 31T^{2} \)
37 \( 1 - 11.1T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 - 6.46T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 4.19T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 5.66T + 73T^{2} \)
79 \( 1 - 6.19T + 79T^{2} \)
83 \( 1 + 2.19T + 83T^{2} \)
89 \( 1 - 17.1T + 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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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

−7.80366694230644971267200281217, −7.48055651906683752018593183982, −6.80641808270507551526578347551, −5.77179584881438862264453518382, −4.88795858040285611245616278232, −4.00976060686471117412418566200, −3.37401466815977721440861306340, −2.48640336881630529349935287864, −1.77597565448545206966358290055, −1.05523977213570428593308749619, 1.05523977213570428593308749619, 1.77597565448545206966358290055, 2.48640336881630529349935287864, 3.37401466815977721440861306340, 4.00976060686471117412418566200, 4.88795858040285611245616278232, 5.77179584881438862264453518382, 6.80641808270507551526578347551, 7.48055651906683752018593183982, 7.80366694230644971267200281217

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