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Properties

Label 1-2624-2624.1349-r0-0-0
Degree $1$
Conductor $2624$
Sign $-0.768 - 0.640i$
Analytic cond. $12.1858$
Root an. cond. $12.1858$
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)3-s + (−0.522 − 0.852i)5-s + (0.453 − 0.891i)7-s + (−0.707 + 0.707i)9-s + (−0.233 − 0.972i)11-s + (−0.0784 − 0.996i)13-s + (0.587 − 0.809i)15-s + (0.587 + 0.809i)17-s + (−0.649 − 0.760i)19-s + (0.996 + 0.0784i)21-s + (−0.453 − 0.891i)23-s + (−0.453 + 0.891i)25-s + (−0.923 − 0.382i)27-s + (−0.233 + 0.972i)29-s + (0.809 − 0.587i)31-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)3-s + (−0.522 − 0.852i)5-s + (0.453 − 0.891i)7-s + (−0.707 + 0.707i)9-s + (−0.233 − 0.972i)11-s + (−0.0784 − 0.996i)13-s + (0.587 − 0.809i)15-s + (0.587 + 0.809i)17-s + (−0.649 − 0.760i)19-s + (0.996 + 0.0784i)21-s + (−0.453 − 0.891i)23-s + (−0.453 + 0.891i)25-s + (−0.923 − 0.382i)27-s + (−0.233 + 0.972i)29-s + (0.809 − 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $-0.768 - 0.640i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ -0.768 - 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2850001228 - 0.7871875834i\)
\(L(\frac12)\) \(\approx\) \(0.2850001228 - 0.7871875834i\)
\(L(1)\) \(\approx\) \(0.9378086574 - 0.1432880982i\)
\(L(1)\) \(\approx\) \(0.9378086574 - 0.1432880982i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (-0.522 - 0.852i)T \)
7 \( 1 + (0.453 - 0.891i)T \)
11 \( 1 + (-0.233 - 0.972i)T \)
13 \( 1 + (-0.0784 - 0.996i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (-0.649 - 0.760i)T \)
23 \( 1 + (-0.453 - 0.891i)T \)
29 \( 1 + (-0.233 + 0.972i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.522 + 0.852i)T \)
43 \( 1 + (-0.760 - 0.649i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (0.852 - 0.522i)T \)
59 \( 1 + (0.649 - 0.760i)T \)
61 \( 1 + (-0.760 + 0.649i)T \)
67 \( 1 + (0.233 - 0.972i)T \)
71 \( 1 + (0.987 + 0.156i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.923 + 0.382i)T \)
89 \( 1 + (-0.453 + 0.891i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−19.34728679243473749221026765830, −18.973945952785169725744578456106, −18.17730629205807660281456649042, −17.93254241512809120080053121450, −16.938774256152963919747308148040, −15.89506551639290313814027638383, −15.20176616099956250819455765456, −14.53598084484262194105887072771, −14.16304178992869331088149189815, −13.20107487571435643433819708030, −12.29025497321258444494912538188, −11.77916560850157828069392532891, −11.39802550961188727770262595092, −10.136970606291793955040545006077, −9.461619072544220637248891893348, −8.55536502744699470197436503503, −7.76311700718418315739612481076, −7.366708487341245576450532490737, −6.47033941863077343829167295828, −5.84389153329128066510596774742, −4.72332591247714245339818365797, −3.81532226566200950137484326028, −2.81584410851932922335005718670, −2.19994762618370895816360608363, −1.48590721172880894441041450986, 0.25105673801055294202337149379, 1.24363762156671323290502030155, 2.57239031885687419430476410908, 3.52959411913304499176696349470, 4.04308765739304942203833859084, 4.914460631821897175031220079426, 5.41210019670001269048131365458, 6.51525396368893410082832391113, 7.75722905690664103483737556807, 8.32881275965265193987568583008, 8.58593822149901085525439603762, 9.786875351539082453290079313192, 10.4209946125588040629608469337, 11.01342527713812184685247682363, 11.73069327876830409583980389655, 12.8356778700361518910660089278, 13.343757365601736437619843395546, 14.16798694326102814872294353709, 14.96888283528993438752816012094, 15.46240800448933006212197993357, 16.3884395231491791627706473691, 16.78107169583208810125763646230, 17.3296253356377382730725638871, 18.42572315392537787205180639316, 19.3961020046018647345712550400

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