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Properties

Label 1-5915-5915.2078-r1-0-0
Degree $1$
Conductor $5915$
Sign $-0.587 - 0.809i$
Analytic cond. $635.655$
Root an. cond. $635.655$
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  + (0.692 − 0.721i)2-s + (0.600 − 0.799i)3-s + (−0.0402 − 0.999i)4-s + (−0.160 − 0.987i)6-s + (−0.748 − 0.663i)8-s + (−0.278 − 0.960i)9-s + (−0.960 − 0.278i)11-s + (−0.822 − 0.568i)12-s + (−0.996 + 0.0804i)16-s + (0.979 + 0.200i)17-s + (−0.885 − 0.464i)18-s + (0.866 − 0.5i)19-s + (−0.866 + 0.5i)22-s + (0.866 + 0.5i)23-s + (−0.979 + 0.200i)24-s + ⋯
L(s)  = 1  + (0.692 − 0.721i)2-s + (0.600 − 0.799i)3-s + (−0.0402 − 0.999i)4-s + (−0.160 − 0.987i)6-s + (−0.748 − 0.663i)8-s + (−0.278 − 0.960i)9-s + (−0.960 − 0.278i)11-s + (−0.822 − 0.568i)12-s + (−0.996 + 0.0804i)16-s + (0.979 + 0.200i)17-s + (−0.885 − 0.464i)18-s + (0.866 − 0.5i)19-s + (−0.866 + 0.5i)22-s + (0.866 + 0.5i)23-s + (−0.979 + 0.200i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5915\)    =    \(5 \cdot 7 \cdot 13^{2}\)
Sign: $-0.587 - 0.809i$
Analytic conductor: \(635.655\)
Root analytic conductor: \(635.655\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5915} (2078, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5915,\ (1:\ ),\ -0.587 - 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.204664251 - 4.324828470i\)
\(L(\frac12)\) \(\approx\) \(2.204664251 - 4.324828470i\)
\(L(1)\) \(\approx\) \(1.346308096 - 1.317224077i\)
\(L(1)\) \(\approx\) \(1.346308096 - 1.317224077i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.692 - 0.721i)T \)
3 \( 1 + (0.600 - 0.799i)T \)
11 \( 1 + (-0.960 - 0.278i)T \)
17 \( 1 + (0.979 + 0.200i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.692 + 0.721i)T \)
31 \( 1 + (0.935 + 0.354i)T \)
37 \( 1 + (0.632 + 0.774i)T \)
41 \( 1 + (0.600 - 0.799i)T \)
43 \( 1 + (0.774 + 0.632i)T \)
47 \( 1 + (0.885 - 0.464i)T \)
53 \( 1 + (0.663 - 0.748i)T \)
59 \( 1 + (-0.0804 + 0.996i)T \)
61 \( 1 + (-0.948 - 0.316i)T \)
67 \( 1 + (-0.0402 + 0.999i)T \)
71 \( 1 + (-0.391 - 0.919i)T \)
73 \( 1 + (0.970 - 0.239i)T \)
79 \( 1 + (-0.885 + 0.464i)T \)
83 \( 1 + (0.120 + 0.992i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.428 + 0.903i)T \)
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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

−17.573591246696533381175589472082, −16.82644598600065721997116638709, −16.39706078110606571201607604573, −15.58888002418002864523539901442, −15.35981316425446960846096989251, −14.4597962485377756846105650309, −14.12537754781562563295512597064, −13.34531904773049768420969116728, −12.77905553468177287875618157084, −12.00346577269259329494894598476, −11.2075958977577677979412697438, −10.46246391739275354532314120508, −9.67957879769132295191466895221, −9.117416394508798277148862559, −8.24245964355625180405109737409, −7.626535918860184883330710748470, −7.322420184895636718183758442133, −5.949087640322800310248453158904, −5.60671962796441059172426099148, −4.734666183508593045267648872017, −4.28279665262731535475466347042, −3.34007716890964245209796290892, −2.84656116053185827035773777937, −2.1196495604983576939294791133, −0.629090145841819101122939331893, 0.721756378378504909969809301164, 1.11759128030786856263550483797, 2.10598854112299628238841711484, 2.90941996880216023883312592168, 3.21083461745619280658320085030, 4.12578353735018353501995233752, 5.177733697086521009112171921741, 5.60381034426353384825229844663, 6.450208495072722919239825000204, 7.29867063022706610444269860055, 7.80033901752666040058856199359, 8.785000697668626089892983145845, 9.35319261184027541007374430804, 10.124432872022187548627030301290, 10.834843526867415193417684144562, 11.5870024935829842636224558874, 12.17752853630067195690696080838, 12.83988811994203676431550293779, 13.369861137613123776647730954294, 13.871747335817027134585345546765, 14.527733385757855676266110016179, 15.20552804864726773814636553703, 15.73762430237072397653793011258, 16.67111721996510179601192748245, 17.64218917069616589436032449704

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