Properties

Label 2-349-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.107·2-s − 2.82·3-s − 1.98·4-s − 3.72·5-s + 0.303·6-s − 3.10·7-s + 0.429·8-s + 4.97·9-s + 0.401·10-s + 4.91·11-s + 5.61·12-s − 2.18·13-s + 0.334·14-s + 10.5·15-s + 3.93·16-s − 4.77·17-s − 0.535·18-s + 0.688·19-s + 7.41·20-s + 8.76·21-s − 0.528·22-s − 7.77·23-s − 1.21·24-s + 8.88·25-s + 0.235·26-s − 5.58·27-s + 6.17·28-s + ⋯
L(s)  = 1  − 0.0760·2-s − 1.63·3-s − 0.994·4-s − 1.66·5-s + 0.124·6-s − 1.17·7-s + 0.151·8-s + 1.65·9-s + 0.126·10-s + 1.48·11-s + 1.62·12-s − 0.606·13-s + 0.0892·14-s + 2.71·15-s + 0.982·16-s − 1.15·17-s − 0.126·18-s + 0.157·19-s + 1.65·20-s + 1.91·21-s − 0.112·22-s − 1.62·23-s − 0.247·24-s + 1.77·25-s + 0.0461·26-s − 1.07·27-s + 1.16·28-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: \(0\)
Selberg data: \((2,\ 349,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1895690888\)
\(L(\frac12)\) \(\approx\) \(0.1895690888\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.107T + 2T^{2} \)
3 \( 1 + 2.82T + 3T^{2} \)
5 \( 1 + 3.72T + 5T^{2} \)
7 \( 1 + 3.10T + 7T^{2} \)
11 \( 1 - 4.91T + 11T^{2} \)
13 \( 1 + 2.18T + 13T^{2} \)
17 \( 1 + 4.77T + 17T^{2} \)
19 \( 1 - 0.688T + 19T^{2} \)
23 \( 1 + 7.77T + 23T^{2} \)
29 \( 1 + 2.94T + 29T^{2} \)
31 \( 1 + 5.80T + 31T^{2} \)
37 \( 1 + 3.19T + 37T^{2} \)
41 \( 1 - 7.23T + 41T^{2} \)
43 \( 1 - 6.47T + 43T^{2} \)
47 \( 1 - 9.63T + 47T^{2} \)
53 \( 1 - 6.04T + 53T^{2} \)
59 \( 1 - 5.58T + 59T^{2} \)
61 \( 1 + 0.633T + 61T^{2} \)
67 \( 1 - 0.475T + 67T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 + 4.37T + 79T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−11.61679101870428245767196961463, −10.77684695565938122613634095144, −9.685368072349372220050404025355, −8.860335175307188788442212388751, −7.47999734892711359449909975023, −6.65848771991521861122392226046, −5.62344741431607035308173301140, −4.21053403176283385368831582718, −3.93320622045546731710510617968, −0.45743924651590474213246576399, 0.45743924651590474213246576399, 3.93320622045546731710510617968, 4.21053403176283385368831582718, 5.62344741431607035308173301140, 6.65848771991521861122392226046, 7.47999734892711359449909975023, 8.860335175307188788442212388751, 9.685368072349372220050404025355, 10.77684695565938122613634095144, 11.61679101870428245767196961463

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