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Properties

Label 1-2624-2624.1083-r0-0-0
Degree $1$
Conductor $2624$
Sign $-0.955 + 0.293i$
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  + (0.923 + 0.382i)3-s + (0.233 + 0.972i)5-s + (−0.309 + 0.951i)7-s + (0.707 + 0.707i)9-s + (−0.233 + 0.972i)11-s + (−0.996 − 0.0784i)13-s + (−0.156 + 0.987i)15-s + (0.987 − 0.156i)17-s + (−0.649 + 0.760i)19-s + (−0.649 + 0.760i)21-s + (0.891 + 0.453i)23-s + (−0.891 + 0.453i)25-s + (0.382 + 0.923i)27-s + (−0.972 + 0.233i)29-s + (0.809 + 0.587i)31-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)3-s + (0.233 + 0.972i)5-s + (−0.309 + 0.951i)7-s + (0.707 + 0.707i)9-s + (−0.233 + 0.972i)11-s + (−0.996 − 0.0784i)13-s + (−0.156 + 0.987i)15-s + (0.987 − 0.156i)17-s + (−0.649 + 0.760i)19-s + (−0.649 + 0.760i)21-s + (0.891 + 0.453i)23-s + (−0.891 + 0.453i)25-s + (0.382 + 0.923i)27-s + (−0.972 + 0.233i)29-s + (0.809 + 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $-0.955 + 0.293i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1083, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ -0.955 + 0.293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3097485326 + 2.064580697i\)
\(L(\frac12)\) \(\approx\) \(0.3097485326 + 2.064580697i\)
\(L(1)\) \(\approx\) \(1.126784186 + 0.8157641292i\)
\(L(1)\) \(\approx\) \(1.126784186 + 0.8157641292i\)

Euler product

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

−19.104794892640093325737831597255, −18.62878536325703146643437709496, −17.356068731922756962882325398691, −16.97194701326175520321103796704, −16.33524576814549183465223885438, −15.4248526378261857235173849531, −14.61206411927209738808532568565, −13.981546663208877387149207282087, −13.107823642225354971437569913531, −13.01800647832617557769340895736, −12.06715478772807448152927930107, −11.10147640047987261361997332525, −10.11458380978082985571232354925, −9.52907556475473986896665994660, −8.85023546041212145566302119172, −8.02261551062232648868040427375, −7.53468445090809829601902968703, −6.61850353443423425338371155417, −5.7715165690164357433404400108, −4.69459745951360511103495645882, −4.094797971345364356607887622657, −3.10005927047956682989304930777, −2.40299998815447180284126532902, −1.22331833537958274201405521868, −0.57566053243696738640089999814, 1.72169786417983494357724598000, 2.38804344029692481161937540977, 2.99812142751151130387345764345, 3.752918827725223518764930420802, 4.84410473106247221095936070846, 5.542692136529155363529661922812, 6.55230692351801101526392540832, 7.43376573865590514823104397882, 7.83933943561429841754715817500, 8.97384531991723881118087531521, 9.61229780426501679658749444609, 10.09195165505830986445285004908, 10.8015549198210237913063093311, 11.91973992603057860282617304203, 12.572758767445968336950236637781, 13.26578173615265101834342455942, 14.34289805475142489639226182818, 14.70563888827515347087947198287, 15.20462648032333352091939697730, 15.85696735509602974973827099440, 16.85108192336976748724253473312, 17.61602263394703357520813512978, 18.56188000551015729968293509376, 18.95390487110423247815416366669, 19.49067335746670971104873005911

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