Properties

Label 2-1368-1.1-c1-0-10
Degree $2$
Conductor $1368$
Sign $1$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
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  + 3.08·5-s + 1.78·7-s + 5.08·11-s − 1.29·13-s − 0.213·17-s − 19-s + 3.72·23-s + 4.51·25-s − 0.870·29-s + 5.51·35-s − 2·37-s − 8.59·41-s − 3.67·43-s + 4.65·47-s − 3.80·49-s − 11.0·53-s + 15.6·55-s + 4.70·59-s + 3.51·61-s − 3.99·65-s + 1.12·67-s − 8.76·71-s + 6.80·73-s + 9.08·77-s + 14.5·79-s + 9.74·83-s − 0.657·85-s + ⋯
L(s)  = 1  + 1.37·5-s + 0.675·7-s + 1.53·11-s − 0.359·13-s − 0.0517·17-s − 0.229·19-s + 0.776·23-s + 0.902·25-s − 0.161·29-s + 0.931·35-s − 0.328·37-s − 1.34·41-s − 0.560·43-s + 0.679·47-s − 0.543·49-s − 1.51·53-s + 2.11·55-s + 0.612·59-s + 0.449·61-s − 0.496·65-s + 0.137·67-s − 1.03·71-s + 0.796·73-s + 1.03·77-s + 1.64·79-s + 1.06·83-s − 0.0713·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.497079525\)
\(L(\frac12)\) \(\approx\) \(2.497079525\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + T \)
good5 \( 1 - 3.08T + 5T^{2} \)
7 \( 1 - 1.78T + 7T^{2} \)
11 \( 1 - 5.08T + 11T^{2} \)
13 \( 1 + 1.29T + 13T^{2} \)
17 \( 1 + 0.213T + 17T^{2} \)
23 \( 1 - 3.72T + 23T^{2} \)
29 \( 1 + 0.870T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 8.59T + 41T^{2} \)
43 \( 1 + 3.67T + 43T^{2} \)
47 \( 1 - 4.65T + 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 - 4.70T + 59T^{2} \)
61 \( 1 - 3.51T + 61T^{2} \)
67 \( 1 - 1.12T + 67T^{2} \)
71 \( 1 + 8.76T + 71T^{2} \)
73 \( 1 - 6.80T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 - 9.74T + 83T^{2} \)
89 \( 1 - 6.76T + 89T^{2} \)
97 \( 1 - 4.16T + 97T^{2} \)
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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

−9.482869884752901656323682465433, −9.004988772190850327614437829251, −8.091828674904306979572205826312, −6.89409903529333011620570340029, −6.38706958702508436414855860757, −5.40032768364559617264722850596, −4.66371073442091522596568126663, −3.46721483379468612275974518197, −2.12920303680536891041040535460, −1.34197692626969631701124950145, 1.34197692626969631701124950145, 2.12920303680536891041040535460, 3.46721483379468612275974518197, 4.66371073442091522596568126663, 5.40032768364559617264722850596, 6.38706958702508436414855860757, 6.89409903529333011620570340029, 8.091828674904306979572205826312, 9.004988772190850327614437829251, 9.482869884752901656323682465433

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