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Properties

Label 1-1344-1344.125-r0-0-0
Degree $1$
Conductor $1344$
Sign $-0.881 + 0.471i$
Analytic cond. $6.24150$
Root an. cond. $6.24150$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.382 + 0.923i)5-s + (−0.923 + 0.382i)11-s + (−0.382 + 0.923i)13-s + i·17-s + (0.382 − 0.923i)19-s + (0.707 + 0.707i)23-s + (−0.707 + 0.707i)25-s + (−0.923 − 0.382i)29-s + 31-s + (−0.382 − 0.923i)37-s + (−0.707 − 0.707i)41-s + (−0.923 + 0.382i)43-s + i·47-s + (−0.923 + 0.382i)53-s + (−0.707 − 0.707i)55-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)5-s + (−0.923 + 0.382i)11-s + (−0.382 + 0.923i)13-s + i·17-s + (0.382 − 0.923i)19-s + (0.707 + 0.707i)23-s + (−0.707 + 0.707i)25-s + (−0.923 − 0.382i)29-s + 31-s + (−0.382 − 0.923i)37-s + (−0.707 − 0.707i)41-s + (−0.923 + 0.382i)43-s + i·47-s + (−0.923 + 0.382i)53-s + (−0.707 − 0.707i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.881 + 0.471i$
Analytic conductor: \(6.24150\)
Root analytic conductor: \(6.24150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1344,\ (0:\ ),\ -0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2246286451 + 0.8967678195i\)
\(L(\frac12)\) \(\approx\) \(0.2246286451 + 0.8967678195i\)
\(L(1)\) \(\approx\) \(0.8726298913 + 0.3391806949i\)
\(L(1)\) \(\approx\) \(0.8726298913 + 0.3391806949i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.382 - 0.923i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (-0.923 - 0.382i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.382 - 0.923i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.923 + 0.382i)T \)
59 \( 1 + (0.382 + 0.923i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 + (-0.923 - 0.382i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.382 + 0.923i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.51516326067901924141676963329, −20.17141517248612041713474704274, −18.89847282842639504534084028811, −18.373873437942401579677025667691, −17.48045840895263485408955987392, −16.74382301302391677454078692660, −16.10786949410321944361206458535, −15.34920562503971456424422638461, −14.41924112491778215192358178896, −13.45177443046189956611889849648, −13.01849013860951407507429187672, −12.17469705178801381288329372206, −11.3832068935375634952652037122, −10.194519010039752679085585546217, −9.859681572182755871590412875998, −8.67131769044959996250383505258, −8.15605327767353902289737251406, −7.244146634552294598975957174292, −6.11624345347094419351848554548, −5.15490460541335636455840451791, −4.9023802745042152449092503923, −3.44237432663368734358076225871, −2.62962908190110842666239254199, −1.46549054313778094052008993843, −0.34126230519117296110287607442, 1.60091979521381417860303226835, 2.44287030750577325549799233216, 3.27324346831712470124298746429, 4.36533808976537869660321327415, 5.31580880305053823267202788492, 6.18724280793624693488355907046, 7.09665205376371368600486122625, 7.602618440037926497999211733940, 8.80323988749934966542818376673, 9.641505818239552765591119389632, 10.34977373109512794842150424782, 11.11324481110024575387377743980, 11.82285618211642922349574235829, 12.95144918134624249754750992038, 13.52706986230544966125383874299, 14.37506062350346537614714878875, 15.157955888611111891252894079879, 15.664312646829128128749515967957, 16.81548998154802294723003826376, 17.54185203917627795364719096798, 18.09912549304716353681457863325, 19.18716782996062190494647895807, 19.29979553220975875240716988832, 20.656173309837071244867701911569, 21.25865113323272256223683937375

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