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Properties

Label 2-1344-48.11-c1-0-40
Degree $2$
Conductor $1344$
Sign $0.412 + 0.910i$
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  + (1.63 − 0.567i)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 + (1.63 − 0.567i)21-s − 6.42i·23-s + 4.96i·25-s + (2.80 − 4.37i)27-s + (5.00 + 5.00i)29-s + 2.52i·31-s + ⋯
L(s)  = 1  + (0.944 − 0.327i)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 + (0.357 − 0.123i)21-s − 1.33i·23-s + 0.992i·25-s + (0.539 − 0.842i)27-s + (0.929 + 0.929i)29-s + 0.454i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.412 + 0.910i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.412 + 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.362228186\)
\(L(\frac12)\) \(\approx\) \(2.362228186\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 + 0.567i)T \)
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

−9.132314394743170694005047241560, −8.785276502170987481318644985399, −7.919412496714000809849784896440, −7.05028418581347366595645955248, −6.46647661022986867908108821874, −5.02351035967864874011084324122, −4.38945884595289244485041494896, −3.00696693777569032778452792177, −2.40160686751984711781519593614, −0.903230767987295186466190793115, 1.66948558685678521446745908808, 2.51022356861378544518247988435, 3.95004851515834631386982722228, 4.22794129137305354902198269055, 5.64787298965087070213051673827, 6.42396180417822076158178300680, 7.73465041699279963227023708171, 8.093113853405502383589431824020, 8.837389670756728613016289880282, 9.832528404855515109436519670330

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