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Properties

Label 1-2404-2404.1027-r0-0-0
Degree $1$
Conductor $2404$
Sign $-0.590 - 0.807i$
Analytic cond. $11.1641$
Root an. cond. $11.1641$
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  + (−0.535 + 0.844i)3-s + i·5-s + (0.509 − 0.860i)7-s + (−0.425 − 0.904i)9-s + (0.827 + 0.562i)11-s + (−0.951 + 0.309i)13-s + (−0.844 − 0.535i)15-s + (−0.987 + 0.156i)17-s + (0.0941 + 0.995i)19-s + (0.453 + 0.891i)21-s + (−0.481 − 0.876i)23-s − 25-s + (0.992 + 0.125i)27-s + (0.995 − 0.0941i)29-s + (−0.960 − 0.278i)31-s + ⋯
L(s)  = 1  + (−0.535 + 0.844i)3-s + i·5-s + (0.509 − 0.860i)7-s + (−0.425 − 0.904i)9-s + (0.827 + 0.562i)11-s + (−0.951 + 0.309i)13-s + (−0.844 − 0.535i)15-s + (−0.987 + 0.156i)17-s + (0.0941 + 0.995i)19-s + (0.453 + 0.891i)21-s + (−0.481 − 0.876i)23-s − 25-s + (0.992 + 0.125i)27-s + (0.995 − 0.0941i)29-s + (−0.960 − 0.278i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2404 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2404 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2404\)    =    \(2^{2} \cdot 601\)
Sign: $-0.590 - 0.807i$
Analytic conductor: \(11.1641\)
Root analytic conductor: \(11.1641\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2404} (1027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2404,\ (0:\ ),\ -0.590 - 0.807i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09067527787 + 0.1786884187i\)
\(L(\frac12)\) \(\approx\) \(-0.09067527787 + 0.1786884187i\)
\(L(1)\) \(\approx\) \(0.6805641387 + 0.3042765604i\)
\(L(1)\) \(\approx\) \(0.6805641387 + 0.3042765604i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
601 \( 1 \)
good3 \( 1 + (-0.535 + 0.844i)T \)
5 \( 1 + iT \)
7 \( 1 + (0.509 - 0.860i)T \)
11 \( 1 + (0.827 + 0.562i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (-0.987 + 0.156i)T \)
19 \( 1 + (0.0941 + 0.995i)T \)
23 \( 1 + (-0.481 - 0.876i)T \)
29 \( 1 + (0.995 - 0.0941i)T \)
31 \( 1 + (-0.960 - 0.278i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (0.975 - 0.218i)T \)
43 \( 1 + (-0.940 + 0.338i)T \)
47 \( 1 + (0.992 + 0.125i)T \)
53 \( 1 + (-0.338 + 0.940i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.248 - 0.968i)T \)
67 \( 1 + (-0.968 - 0.248i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + (0.397 - 0.917i)T \)
79 \( 1 + (-0.750 + 0.661i)T \)
83 \( 1 + (-0.612 + 0.790i)T \)
89 \( 1 + (0.992 - 0.125i)T \)
97 \( 1 + (-0.453 - 0.891i)T \)
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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

−19.26870286290049028824526219293, −18.09066994616705028990834730864, −17.67455462576696275949059389461, −17.163202598544697115143856408847, −16.26711308675689366062696839965, −15.68000080288041247267205986134, −14.71123626946323282578926525523, −13.87435900112904420561410424577, −13.208303686362875884694691847923, −12.47365523471886566850721141600, −11.84027375049574766939792768332, −11.457821138586804159717905702011, −10.493483419085931234956106683, −9.09809703836165559781173919811, −8.977282439444104967439187472014, −7.9785338423404520982803330118, −7.26607593636157537668598652777, −6.34979003775632562882935211066, −5.53695152052435612194253207150, −5.03553999588742730118683731369, −4.202414279503281723412894364132, −2.78346290932677557509715882180, −1.982026152833795729536089601254, −1.188240468291155314525059124904, −0.07122683870129589689215090881, 1.48855046941212052046978927960, 2.49798830975188946979361179002, 3.583828721340451238868457676661, 4.292725100274531672301083765548, 4.74727209479679965457950157717, 5.99459254636747523509159043346, 6.63669521614832820174276558901, 7.27075724829539984849045702623, 8.2171968901338388643651371538, 9.30531796731397938153587210629, 9.98447991053700116069393303463, 10.53311713200664532410163485968, 11.1695734406696300671551626716, 11.892011381083152176323801880621, 12.544559248806581918078973121930, 13.97368144619091314140904423781, 14.30790424841926954283591980109, 14.956048146092913924899182954880, 15.61000676424559442965394728631, 16.65130491822225792981636651726, 17.08459021988611449002517218461, 17.75100799775924632393047081416, 18.3718048102229305180639792033, 19.408138209955868692788047313334, 20.14165062242963087767670797819

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