Properties

Label 2-349-1.1-c1-0-27
Degree $2$
Conductor $349$
Sign $-1$
Analytic cond. $2.78677$
Root an. cond. $1.66936$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.91·2-s − 2.24·3-s + 1.67·4-s − 1.23·5-s − 4.30·6-s − 3.37·7-s − 0.631·8-s + 2.05·9-s − 2.37·10-s − 1.90·11-s − 3.75·12-s − 0.969·13-s − 6.46·14-s + 2.78·15-s − 4.55·16-s + 7.23·17-s + 3.94·18-s − 3.38·19-s − 2.06·20-s + 7.58·21-s − 3.65·22-s + 0.502·23-s + 1.42·24-s − 3.46·25-s − 1.85·26-s + 2.11·27-s − 5.63·28-s + ⋯
L(s)  = 1  + 1.35·2-s − 1.29·3-s + 0.835·4-s − 0.553·5-s − 1.75·6-s − 1.27·7-s − 0.223·8-s + 0.686·9-s − 0.749·10-s − 0.575·11-s − 1.08·12-s − 0.268·13-s − 1.72·14-s + 0.718·15-s − 1.13·16-s + 1.75·17-s + 0.929·18-s − 0.777·19-s − 0.462·20-s + 1.65·21-s − 0.779·22-s + 0.104·23-s + 0.290·24-s − 0.693·25-s − 0.364·26-s + 0.407·27-s − 1.06·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $-1$
Analytic conductor: \(2.78677\)
Root analytic conductor: \(1.66936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 349,\ (\ :1/2),\ -1)\)

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)$
bad349 \( 1 + T \)
good2 \( 1 - 1.91T + 2T^{2} \)
3 \( 1 + 2.24T + 3T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 + 3.37T + 7T^{2} \)
11 \( 1 + 1.90T + 11T^{2} \)
13 \( 1 + 0.969T + 13T^{2} \)
17 \( 1 - 7.23T + 17T^{2} \)
19 \( 1 + 3.38T + 19T^{2} \)
23 \( 1 - 0.502T + 23T^{2} \)
29 \( 1 + 7.43T + 29T^{2} \)
31 \( 1 - 3.69T + 31T^{2} \)
37 \( 1 - 4.21T + 37T^{2} \)
41 \( 1 - 5.83T + 41T^{2} \)
43 \( 1 + 7.91T + 43T^{2} \)
47 \( 1 + 5.52T + 47T^{2} \)
53 \( 1 - 0.948T + 53T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 + 1.46T + 67T^{2} \)
71 \( 1 + 3.08T + 71T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 - 4.39T + 79T^{2} \)
83 \( 1 - 7.14T + 83T^{2} \)
89 \( 1 - 2.83T + 89T^{2} \)
97 \( 1 - 8.19T + 97T^{2} \)
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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

−11.46586720143260920857737800469, −10.39878121050227667636689361145, −9.491448552674230298479859768841, −7.84399533687752038321416620869, −6.64864731095534240394190747658, −5.89672601431754534870545784780, −5.20001821010109954039075110070, −4.01448683881000773533878028347, −3.00395971167089935508228267065, 0, 3.00395971167089935508228267065, 4.01448683881000773533878028347, 5.20001821010109954039075110070, 5.89672601431754534870545784780, 6.64864731095534240394190747658, 7.84399533687752038321416620869, 9.491448552674230298479859768841, 10.39878121050227667636689361145, 11.46586720143260920857737800469

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