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Properties

Label 2-1344-112.27-c1-0-4
Degree $2$
Conductor $1344$
Sign $-0.724 + 0.688i$
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 + (−2.49 + 2.49i)5-s + (−0.134 + 2.64i)7-s − 1.00i·9-s + (−3.74 + 3.74i)11-s + (3.44 + 3.44i)13-s − 3.52i·15-s − 0.279i·17-s + (−2.51 + 2.51i)19-s + (−1.77 − 1.96i)21-s − 3.90·23-s − 7.45i·25-s + (0.707 + 0.707i)27-s + (5.23 − 5.23i)29-s + 6.11·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−1.11 + 1.11i)5-s + (−0.0507 + 0.998i)7-s − 0.333i·9-s + (−1.12 + 1.12i)11-s + (0.955 + 0.955i)13-s − 0.911i·15-s − 0.0677i·17-s + (−0.577 + 0.577i)19-s + (−0.386 − 0.428i)21-s − 0.814·23-s − 1.49i·25-s + (0.136 + 0.136i)27-s + (0.971 − 0.971i)29-s + 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.724 + 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.724 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.724 + 0.688i$
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: \((2,\ 1344,\ (\ :1/2),\ -0.724 + 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5947911555\)
\(L(\frac12)\) \(\approx\) \(0.5947911555\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.134 - 2.64i)T \)
good5 \( 1 + (2.49 - 2.49i)T - 5iT^{2} \)
11 \( 1 + (3.74 - 3.74i)T - 11iT^{2} \)
13 \( 1 + (-3.44 - 3.44i)T + 13iT^{2} \)
17 \( 1 + 0.279iT - 17T^{2} \)
19 \( 1 + (2.51 - 2.51i)T - 19iT^{2} \)
23 \( 1 + 3.90T + 23T^{2} \)
29 \( 1 + (-5.23 + 5.23i)T - 29iT^{2} \)
31 \( 1 - 6.11T + 31T^{2} \)
37 \( 1 + (-6.48 - 6.48i)T + 37iT^{2} \)
41 \( 1 + 1.02T + 41T^{2} \)
43 \( 1 + (2.37 - 2.37i)T - 43iT^{2} \)
47 \( 1 + 5.55T + 47T^{2} \)
53 \( 1 + (3.03 + 3.03i)T + 53iT^{2} \)
59 \( 1 + (1.85 + 1.85i)T + 59iT^{2} \)
61 \( 1 + (7.02 + 7.02i)T + 61iT^{2} \)
67 \( 1 + (-0.678 - 0.678i)T + 67iT^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 - 5.13T + 73T^{2} \)
79 \( 1 + 12.7iT - 79T^{2} \)
83 \( 1 + (-1.28 + 1.28i)T - 83iT^{2} \)
89 \( 1 + 2.78T + 89T^{2} \)
97 \( 1 - 13.3iT - 97T^{2} \)
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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

−10.16740415078756787491652406754, −9.515763662583041300570592455016, −8.158122924822257907875711672971, −7.957999881665385841492137676182, −6.53685473332573433509463257856, −6.31279027498582308836579819912, −4.89986395167126924334490742344, −4.19567037810899528018128952896, −3.14172691425274929406230512950, −2.15528375847076997313872357921, 0.31381719165585209130126503702, 1.04251781606754428686880644316, 3.00406095056249854793515124456, 4.00187085438439519276249723240, 4.83623970365206845511696240687, 5.71101321311705730099227002142, 6.65543716519011962677638438218, 7.82856895818752768001740092011, 8.110982902674296063379696997924, 8.750637311471916022878642517396

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