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Properties

Label 2-1344-48.35-c1-0-7
Degree $2$
Conductor $1344$
Sign $0.351 - 0.936i$
Analytic cond. $10.7318$
Root an. cond. $3.27595$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.277 − 1.70i)3-s + (0.459 + 0.459i)5-s − 7-s + (−2.84 + 0.947i)9-s + (−1.43 + 1.43i)11-s + (−1.18 − 1.18i)13-s + (0.658 − 0.912i)15-s + 7.20i·17-s + (2.23 − 2.23i)19-s + (0.277 + 1.70i)21-s + 0.540i·23-s − 4.57i·25-s + (2.40 + 4.60i)27-s + (−4.14 + 4.14i)29-s + 10.4i·31-s + ⋯
L(s)  = 1  + (−0.159 − 0.987i)3-s + (0.205 + 0.205i)5-s − 0.377·7-s + (−0.948 + 0.315i)9-s + (−0.432 + 0.432i)11-s + (−0.328 − 0.328i)13-s + (0.169 − 0.235i)15-s + 1.74i·17-s + (0.512 − 0.512i)19-s + (0.0604 + 0.373i)21-s + 0.112i·23-s − 0.915i·25-s + (0.463 + 0.886i)27-s + (−0.769 + 0.769i)29-s + 1.87i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.351 - 0.936i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.351 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8562808877\)
\(L(\frac12)\) \(\approx\) \(0.8562808877\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.277 + 1.70i)T \)
7 \( 1 + T \)
good5 \( 1 + (-0.459 - 0.459i)T + 5iT^{2} \)
11 \( 1 + (1.43 - 1.43i)T - 11iT^{2} \)
13 \( 1 + (1.18 + 1.18i)T + 13iT^{2} \)
17 \( 1 - 7.20iT - 17T^{2} \)
19 \( 1 + (-2.23 + 2.23i)T - 19iT^{2} \)
23 \( 1 - 0.540iT - 23T^{2} \)
29 \( 1 + (4.14 - 4.14i)T - 29iT^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 + (1.36 - 1.36i)T - 37iT^{2} \)
41 \( 1 + 2.16T + 41T^{2} \)
43 \( 1 + (-6.40 - 6.40i)T + 43iT^{2} \)
47 \( 1 + 7.63T + 47T^{2} \)
53 \( 1 + (-2.29 - 2.29i)T + 53iT^{2} \)
59 \( 1 + (-6.32 + 6.32i)T - 59iT^{2} \)
61 \( 1 + (-0.637 - 0.637i)T + 61iT^{2} \)
67 \( 1 + (-2.14 + 2.14i)T - 67iT^{2} \)
71 \( 1 - 14.6iT - 71T^{2} \)
73 \( 1 - 0.233iT - 73T^{2} \)
79 \( 1 + 4.93iT - 79T^{2} \)
83 \( 1 + (1.76 + 1.76i)T + 83iT^{2} \)
89 \( 1 + 7.79T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−9.883027789505703847299710446715, −8.751560485844781961490610884956, −8.116264521728420736199945358119, −7.21142975372725341791062421566, −6.58583643349903240726755665208, −5.75573138360582730102079704006, −4.90489800041949210813005917101, −3.47436184367037745850404906325, −2.49415165483447635360449048175, −1.38894075149297121902003459853, 0.35998077027566232532996319656, 2.40088894795459474681644698861, 3.38859074064925225643138671489, 4.33436375084532774464389709163, 5.34643887390025088150961835362, 5.79622804160907717733964685178, 7.03014624545493009211956649945, 7.87825554451181177313781396602, 8.945921361197171430869727655635, 9.600064329399478749881618001354

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