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Properties

Label 1-5915-5915.74-r0-0-0
Degree $1$
Conductor $5915$
Sign $-0.0942 - 0.995i$
Analytic cond. $27.4691$
Root an. cond. $27.4691$
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  + (0.354 − 0.935i)2-s + (0.996 − 0.0804i)3-s + (−0.748 − 0.663i)4-s + (0.278 − 0.960i)6-s + (−0.885 + 0.464i)8-s + (0.987 − 0.160i)9-s + (0.987 + 0.160i)11-s + (−0.799 − 0.600i)12-s + (0.120 + 0.992i)16-s + (−0.885 + 0.464i)17-s + (0.200 − 0.979i)18-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)22-s − 23-s + (−0.845 + 0.534i)24-s + ⋯
L(s)  = 1  + (0.354 − 0.935i)2-s + (0.996 − 0.0804i)3-s + (−0.748 − 0.663i)4-s + (0.278 − 0.960i)6-s + (−0.885 + 0.464i)8-s + (0.987 − 0.160i)9-s + (0.987 + 0.160i)11-s + (−0.799 − 0.600i)12-s + (0.120 + 0.992i)16-s + (−0.885 + 0.464i)17-s + (0.200 − 0.979i)18-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)22-s − 23-s + (−0.845 + 0.534i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0942 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0942 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5915\)    =    \(5 \cdot 7 \cdot 13^{2}\)
Sign: $-0.0942 - 0.995i$
Analytic conductor: \(27.4691\)
Root analytic conductor: \(27.4691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5915} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5915,\ (0:\ ),\ -0.0942 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.124885259 - 2.335475500i\)
\(L(\frac12)\) \(\approx\) \(2.124885259 - 2.335475500i\)
\(L(1)\) \(\approx\) \(1.458180493 - 0.9123694147i\)
\(L(1)\) \(\approx\) \(1.458180493 - 0.9123694147i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.354 - 0.935i)T \)
3 \( 1 + (0.996 - 0.0804i)T \)
11 \( 1 + (0.987 + 0.160i)T \)
17 \( 1 + (-0.885 + 0.464i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (0.987 - 0.160i)T \)
31 \( 1 + (0.692 + 0.721i)T \)
37 \( 1 + (0.970 - 0.239i)T \)
41 \( 1 + (0.428 - 0.903i)T \)
43 \( 1 + (-0.692 + 0.721i)T \)
47 \( 1 + (0.200 + 0.979i)T \)
53 \( 1 + (0.845 + 0.534i)T \)
59 \( 1 + (0.120 - 0.992i)T \)
61 \( 1 + (-0.0402 - 0.999i)T \)
67 \( 1 + (0.200 + 0.979i)T \)
71 \( 1 + (-0.996 + 0.0804i)T \)
73 \( 1 + (0.632 - 0.774i)T \)
79 \( 1 + (-0.200 - 0.979i)T \)
83 \( 1 + (-0.568 + 0.822i)T \)
89 \( 1 + T \)
97 \( 1 + (0.919 + 0.391i)T \)
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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

−18.03811460280055085880967114862, −17.0033213452706404305927573164, −16.528489199583655905208832367481, −15.826268031119326498718078882240, −15.20449594900970627388923442877, −14.63988124425874396410051725449, −14.097995110317281404125879299159, −13.51317010640625985862959555257, −12.99596166222497685775696417067, −12.069620679910313978767677359875, −11.60847064713961129273764987473, −10.267431676383178392778583441635, −9.78769444658378556220391816876, −8.9434385078258518658141838455, −8.51384178182809266036463923692, −7.901050117976303706398027236906, −7.12669667158790717535554310284, −6.47072563748600555532769161472, −5.913172078342580265378550694360, −4.76375106081645550010940234871, −4.20343568795771405115081369529, −3.705134503097925528065864174808, −2.79643038133910909993932370196, −2.0263293536800245492241693104, −0.87235101956248788029070948359, 0.75502129783942758319839212010, 1.63165046703789016179502944151, 2.29429969665450071266561664902, 2.91857560379300744247231022762, 3.81312179221213504099251627977, 4.32376377594666476676672308225, 4.868714772574492896945800738139, 6.229085319637477532974561755896, 6.54124233415407189569597136185, 7.632699784612168128233735750920, 8.524466382367770915629118544935, 8.91205330158954295214319211604, 9.594429271665172782535721837462, 10.23897563341003247568265460496, 10.92986158646608470846656923760, 11.73591060606554772364006461445, 12.34520422976141638317166841952, 13.023944262836899545474789932896, 13.56125077706552009641265494372, 14.293854857976845477142384558429, 14.60584584262996222137431010445, 15.484483681102846363074830755, 15.9208785564744439336416186912, 17.21880424385273510667576520233, 17.717287783131099360410408173621

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