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Properties

Label 2-1344-48.35-c1-0-43
Degree $2$
Conductor $1344$
Sign $-0.970 + 0.239i$
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  + (−0.567 − 1.63i)3-s + (−0.132 − 0.132i)5-s + 7-s + (−2.35 + 1.85i)9-s + (0.715 − 0.715i)11-s + (0.206 + 0.206i)13-s + (−0.142 + 0.292i)15-s − 6.91i·17-s + (−5.87 + 5.87i)19-s + (−0.567 − 1.63i)21-s − 6.42i·23-s − 4.96i·25-s + (4.37 + 2.80i)27-s + (−5.00 + 5.00i)29-s − 2.52i·31-s + ⋯
L(s)  = 1  + (−0.327 − 0.944i)3-s + (−0.0594 − 0.0594i)5-s + 0.377·7-s + (−0.785 + 0.618i)9-s + (0.215 − 0.215i)11-s + (0.0572 + 0.0572i)13-s + (−0.0366 + 0.0755i)15-s − 1.67i·17-s + (−1.34 + 1.34i)19-s + (−0.123 − 0.357i)21-s − 1.33i·23-s − 0.992i·25-s + (0.842 + 0.539i)27-s + (−0.929 + 0.929i)29-s − 0.454i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.970 + 0.239i$
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: \((2,\ 1344,\ (\ :1/2),\ -0.970 + 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8034333617\)
\(L(\frac12)\) \(\approx\) \(0.8034333617\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.567 + 1.63i)T \)
7 \( 1 - T \)
good5 \( 1 + (0.132 + 0.132i)T + 5iT^{2} \)
11 \( 1 + (-0.715 + 0.715i)T - 11iT^{2} \)
13 \( 1 + (-0.206 - 0.206i)T + 13iT^{2} \)
17 \( 1 + 6.91iT - 17T^{2} \)
19 \( 1 + (5.87 - 5.87i)T - 19iT^{2} \)
23 \( 1 + 6.42iT - 23T^{2} \)
29 \( 1 + (5.00 - 5.00i)T - 29iT^{2} \)
31 \( 1 + 2.52iT - 31T^{2} \)
37 \( 1 + (-6.15 + 6.15i)T - 37iT^{2} \)
41 \( 1 + 5.19T + 41T^{2} \)
43 \( 1 + (1.36 + 1.36i)T + 43iT^{2} \)
47 \( 1 + 0.603T + 47T^{2} \)
53 \( 1 + (6.19 + 6.19i)T + 53iT^{2} \)
59 \( 1 + (5.00 - 5.00i)T - 59iT^{2} \)
61 \( 1 + (6.24 + 6.24i)T + 61iT^{2} \)
67 \( 1 + (2.55 - 2.55i)T - 67iT^{2} \)
71 \( 1 + 0.808iT - 71T^{2} \)
73 \( 1 - 11.5iT - 73T^{2} \)
79 \( 1 + 3.48iT - 79T^{2} \)
83 \( 1 + (-0.641 - 0.641i)T + 83iT^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 + 5.56T + 97T^{2} \)
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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

−9.031458365044562779835467218957, −8.316698449239567428590069135687, −7.62800100540611806473836788876, −6.73449382457916765574994789580, −6.06906811642404680543516491971, −5.12096317389249584655008099761, −4.16020683300858010220025987151, −2.74471781028245651231699630503, −1.76758678128407773575531764166, −0.33463860495656245366210798360, 1.68922463630024722299107283199, 3.16405774741027971435995196432, 4.10652154783962790586354899594, 4.79836026870347283516494615382, 5.82022428279918186607475725355, 6.49186721098096189889937765941, 7.65619863789764967621789264840, 8.531292285399118394875982799624, 9.243813381210358978335462053476, 9.970480847142602041517351197079

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