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Properties

Label 2-1344-112.27-c1-0-25
Degree $2$
Conductor $1344$
Sign $-0.747 + 0.664i$
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  + (−0.707 + 0.707i)3-s + (−0.167 + 0.167i)5-s + (2.61 − 0.383i)7-s − 1.00i·9-s + (−2.51 + 2.51i)11-s + (−4.28 − 4.28i)13-s − 0.236i·15-s − 7.14i·17-s + (−3.61 + 3.61i)19-s + (−1.57 + 2.12i)21-s − 5.62·23-s + 4.94i·25-s + (0.707 + 0.707i)27-s + (0.0732 − 0.0732i)29-s + 2.74·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.0748 + 0.0748i)5-s + (0.989 − 0.144i)7-s − 0.333i·9-s + (−0.758 + 0.758i)11-s + (−1.18 − 1.18i)13-s − 0.0611i·15-s − 1.73i·17-s + (−0.830 + 0.830i)19-s + (−0.344 + 0.463i)21-s − 1.17·23-s + 0.988i·25-s + (0.136 + 0.136i)27-s + (0.0135 − 0.0135i)29-s + 0.492·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.747 + 0.664i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.747 + 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3169170173\)
\(L(\frac12)\) \(\approx\) \(0.3169170173\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (-2.61 + 0.383i)T \)
good5 \( 1 + (0.167 - 0.167i)T - 5iT^{2} \)
11 \( 1 + (2.51 - 2.51i)T - 11iT^{2} \)
13 \( 1 + (4.28 + 4.28i)T + 13iT^{2} \)
17 \( 1 + 7.14iT - 17T^{2} \)
19 \( 1 + (3.61 - 3.61i)T - 19iT^{2} \)
23 \( 1 + 5.62T + 23T^{2} \)
29 \( 1 + (-0.0732 + 0.0732i)T - 29iT^{2} \)
31 \( 1 - 2.74T + 31T^{2} \)
37 \( 1 + (0.490 + 0.490i)T + 37iT^{2} \)
41 \( 1 + 9.39T + 41T^{2} \)
43 \( 1 + (-3.30 + 3.30i)T - 43iT^{2} \)
47 \( 1 + 0.799T + 47T^{2} \)
53 \( 1 + (4.68 + 4.68i)T + 53iT^{2} \)
59 \( 1 + (7.78 + 7.78i)T + 59iT^{2} \)
61 \( 1 + (-8.08 - 8.08i)T + 61iT^{2} \)
67 \( 1 + (9.96 + 9.96i)T + 67iT^{2} \)
71 \( 1 + 0.235T + 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 - 0.211iT - 79T^{2} \)
83 \( 1 + (8.74 - 8.74i)T - 83iT^{2} \)
89 \( 1 - 7.54T + 89T^{2} \)
97 \( 1 + 14.4iT - 97T^{2} \)
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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

−9.566699273573772789053167086933, −8.335102340475341041833380357567, −7.65958925216266935516037651583, −7.05170815335542037714444169236, −5.66921183299959358019019410765, −5.04689729482587651475682002224, −4.42490074065659070520306479671, −3.06924116456426015338219257629, −1.95681531995108939806094534174, −0.12795832664501895575943831931, 1.67777432540273526523918928576, 2.53452273935750727409402297105, 4.20481357337663171498183569883, 4.79504448882846927279552656316, 5.87332653193098531468931628919, 6.54209072680475313672179468290, 7.60599787507048382317798330112, 8.266314765969498067058795591659, 8.862393670388170350618416138094, 10.17935167503314985669060397267

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