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Properties

Label 1-1248-1248.701-r1-0-0
Degree $1$
Conductor $1248$
Sign $0.555 - 0.831i$
Analytic cond. $134.116$
Root an. cond. $134.116$
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  + (−0.707 + 0.707i)5-s − i·7-s + (0.707 − 0.707i)11-s + 17-s + (0.707 + 0.707i)19-s − i·23-s − i·25-s + (−0.707 − 0.707i)29-s − 31-s + (−0.707 − 0.707i)35-s + (0.707 − 0.707i)37-s + i·41-s + (0.707 − 0.707i)43-s − 47-s − 49-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)5-s − i·7-s + (0.707 − 0.707i)11-s + 17-s + (0.707 + 0.707i)19-s − i·23-s − i·25-s + (−0.707 − 0.707i)29-s − 31-s + (−0.707 − 0.707i)35-s + (0.707 − 0.707i)37-s + i·41-s + (0.707 − 0.707i)43-s − 47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $0.555 - 0.831i$
Analytic conductor: \(134.116\)
Root analytic conductor: \(134.116\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1248,\ (1:\ ),\ 0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.146760574 - 0.6129562975i\)
\(L(\frac12)\) \(\approx\) \(1.146760574 - 0.6129562975i\)
\(L(1)\) \(\approx\) \(0.9466444485 + 0.09015765200i\)
\(L(1)\) \(\approx\) \(0.9466444485 + 0.09015765200i\)

Euler product

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

−20.716781113742599445322527329943, −20.22714609702183858592671852251, −19.67741832868765432487239177781, −18.90209854522319810370251258957, −17.76718456417005689854336274763, −17.13320320549850412822895448683, −16.43630521258245152301443531840, −15.78445073021413466891586292081, −14.789660245934552323326114158581, −14.14370097244462100275369206640, −13.1180366871258695444049097177, −12.55732152497777598851331692302, −11.57232540927454356843770664881, −11.0788568806748333863232116328, −9.78279301766884229900805380973, −9.3791735170607649421244754474, −8.22273266676820460545336805788, −7.420731952372465649639475097590, −6.959393136895274367505870379492, −5.561645665410268503660674720884, −4.76755789687439716978880015255, −3.900947186524127002777350405467, −3.268838260477509954488529896875, −1.58895205580015162149015453095, −0.925949651167437323051613537271, 0.30924594448573331291033359373, 1.637449918341342446857575154856, 2.847698728386433853402390295363, 3.46547566905402588237171755938, 4.43712825191231075740650471418, 5.743880743654324233966094084597, 6.16282636895270552636223485884, 7.35690583922894948408236105803, 8.01586786376414484671181208380, 8.90678066040038250786748559343, 9.71791246898165763070208711648, 10.72532878144061113741579185729, 11.5309674130638491064255721847, 12.04759422272966649900548533009, 12.84896864308433605239010279453, 14.13939597148207588089822403263, 14.59083313701825115204612353482, 15.28352593259948273844788319144, 16.27761488502727345058610948556, 16.66407627408419693728020222603, 18.03139074000045055223640796250, 18.56992436069522142723842154272, 19.1046378346749161573654778386, 19.86471841983501502515158865178, 20.83916152854053869841111024228

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