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Properties

Label 2-1344-48.35-c1-0-9
Degree $2$
Conductor $1344$
Sign $-0.817 - 0.575i$
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  + (0.119 + 1.72i)3-s + (−0.793 − 0.793i)5-s + 7-s + (−2.97 + 0.414i)9-s + (−0.687 + 0.687i)11-s + (2.91 + 2.91i)13-s + (1.27 − 1.46i)15-s − 1.81i·17-s + (−3.24 + 3.24i)19-s + (0.119 + 1.72i)21-s + 5.92i·23-s − 3.74i·25-s + (−1.07 − 5.08i)27-s + (−1.54 + 1.54i)29-s + 3.60i·31-s + ⋯
L(s)  = 1  + (0.0691 + 0.997i)3-s + (−0.354 − 0.354i)5-s + 0.377·7-s + (−0.990 + 0.138i)9-s + (−0.207 + 0.207i)11-s + (0.807 + 0.807i)13-s + (0.329 − 0.378i)15-s − 0.441i·17-s + (−0.743 + 0.743i)19-s + (0.0261 + 0.377i)21-s + 1.23i·23-s − 0.748i·25-s + (−0.206 − 0.978i)27-s + (−0.287 + 0.287i)29-s + 0.648i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.817 - 0.575i$
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: \((2,\ 1344,\ (\ :1/2),\ -0.817 - 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.081155014\)
\(L(\frac12)\) \(\approx\) \(1.081155014\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.119 - 1.72i)T \)
7 \( 1 - T \)
good5 \( 1 + (0.793 + 0.793i)T + 5iT^{2} \)
11 \( 1 + (0.687 - 0.687i)T - 11iT^{2} \)
13 \( 1 + (-2.91 - 2.91i)T + 13iT^{2} \)
17 \( 1 + 1.81iT - 17T^{2} \)
19 \( 1 + (3.24 - 3.24i)T - 19iT^{2} \)
23 \( 1 - 5.92iT - 23T^{2} \)
29 \( 1 + (1.54 - 1.54i)T - 29iT^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 + (5.10 - 5.10i)T - 37iT^{2} \)
41 \( 1 + 1.19T + 41T^{2} \)
43 \( 1 + (-7.00 - 7.00i)T + 43iT^{2} \)
47 \( 1 + 13.1T + 47T^{2} \)
53 \( 1 + (-2.08 - 2.08i)T + 53iT^{2} \)
59 \( 1 + (0.919 - 0.919i)T - 59iT^{2} \)
61 \( 1 + (-4.21 - 4.21i)T + 61iT^{2} \)
67 \( 1 + (7.87 - 7.87i)T - 67iT^{2} \)
71 \( 1 + 7.99iT - 71T^{2} \)
73 \( 1 + 4.19iT - 73T^{2} \)
79 \( 1 - 8.07iT - 79T^{2} \)
83 \( 1 + (11.0 + 11.0i)T + 83iT^{2} \)
89 \( 1 + 7.73T + 89T^{2} \)
97 \( 1 - 6.61T + 97T^{2} \)
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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

−9.944340569974217702903135834639, −9.074189783040847107770855985058, −8.483241502511664320772089615995, −7.74280468234733031307004016010, −6.55724797223704235782452613823, −5.61365995911201426987779730768, −4.70988851907337362637553736352, −4.06251824380884336915623163385, −3.10764734426520203573034046380, −1.63579080096981387507697538306, 0.43499129146500027229700390776, 1.88994968561955326774280777092, 2.94406456120222448561385285139, 3.95337907033971624322370441550, 5.26637304257136199275127732871, 6.10367599265598808534964427223, 6.87924235627292760454050842222, 7.69256135391365529410408203161, 8.392924363817600843863572436094, 8.929845945058351994351823641932

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