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Properties

Label 2-1344-48.35-c1-0-14
Degree $2$
Conductor $1344$
Sign $0.826 + 0.562i$
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  + (−1.61 − 0.629i)3-s + (−3.08 − 3.08i)5-s + 7-s + (2.20 + 2.03i)9-s + (−1.97 + 1.97i)11-s + (3.27 + 3.27i)13-s + (3.03 + 6.92i)15-s + 2.57i·17-s + (1.18 − 1.18i)19-s + (−1.61 − 0.629i)21-s + 1.21i·23-s + 14.0i·25-s + (−2.28 − 4.66i)27-s + (−1.33 + 1.33i)29-s − 5.84i·31-s + ⋯
L(s)  = 1  + (−0.931 − 0.363i)3-s + (−1.38 − 1.38i)5-s + 0.377·7-s + (0.735 + 0.677i)9-s + (−0.596 + 0.596i)11-s + (0.908 + 0.908i)13-s + (0.784 + 1.78i)15-s + 0.623i·17-s + (0.272 − 0.272i)19-s + (−0.352 − 0.137i)21-s + 0.252i·23-s + 2.81i·25-s + (−0.439 − 0.898i)27-s + (−0.248 + 0.248i)29-s − 1.05i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.826 + 0.562i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.826 + 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8464995992\)
\(L(\frac12)\) \(\approx\) \(0.8464995992\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 + 0.629i)T \)
7 \( 1 - T \)
good5 \( 1 + (3.08 + 3.08i)T + 5iT^{2} \)
11 \( 1 + (1.97 - 1.97i)T - 11iT^{2} \)
13 \( 1 + (-3.27 - 3.27i)T + 13iT^{2} \)
17 \( 1 - 2.57iT - 17T^{2} \)
19 \( 1 + (-1.18 + 1.18i)T - 19iT^{2} \)
23 \( 1 - 1.21iT - 23T^{2} \)
29 \( 1 + (1.33 - 1.33i)T - 29iT^{2} \)
31 \( 1 + 5.84iT - 31T^{2} \)
37 \( 1 + (-5.18 + 5.18i)T - 37iT^{2} \)
41 \( 1 + 0.688T + 41T^{2} \)
43 \( 1 + (0.649 + 0.649i)T + 43iT^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + (-3.84 - 3.84i)T + 53iT^{2} \)
59 \( 1 + (-1.99 + 1.99i)T - 59iT^{2} \)
61 \( 1 + (6.62 + 6.62i)T + 61iT^{2} \)
67 \( 1 + (4.50 - 4.50i)T - 67iT^{2} \)
71 \( 1 - 3.91iT - 71T^{2} \)
73 \( 1 + 9.83iT - 73T^{2} \)
79 \( 1 + 3.86iT - 79T^{2} \)
83 \( 1 + (-2.99 - 2.99i)T + 83iT^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−9.354962140682851572263349772530, −8.656948910093800308549350860995, −7.71305430354421716191981706462, −7.39443064124424414038922526538, −6.10910115167041718394608285853, −5.23105357941951572391672167545, −4.44848862551766560160601441415, −3.89459751705704757533021467800, −1.84111202205138894630790259481, −0.73439240811005077032007308563, 0.68351211990001942850728799729, 2.89005927451454692501821215043, 3.59516224582677622096003789505, 4.52365507024920260369565174455, 5.59591543818279066923546584642, 6.38224296093141895714899393898, 7.27534330324569799555319973964, 7.88156913283692041229656612117, 8.735250931683957566143844785612, 10.18077590572785408819302914242

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