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Properties

Label 1-3751-3751.576-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.931 - 0.364i$
Analytic cond. $17.4195$
Root an. cond. $17.4195$
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  + (−0.959 + 0.281i)2-s + (−0.978 + 0.207i)3-s + (0.841 − 0.540i)4-s + (0.123 − 0.992i)5-s + (0.879 − 0.475i)6-s + (0.580 − 0.814i)7-s + (−0.654 + 0.755i)8-s + (0.913 − 0.406i)9-s + (0.161 + 0.986i)10-s + (−0.710 + 0.703i)12-s + (−0.625 − 0.780i)13-s + (−0.327 + 0.945i)14-s + (0.0855 + 0.996i)15-s + (0.415 − 0.909i)16-s + (0.640 + 0.768i)17-s + (−0.761 + 0.647i)18-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (−0.978 + 0.207i)3-s + (0.841 − 0.540i)4-s + (0.123 − 0.992i)5-s + (0.879 − 0.475i)6-s + (0.580 − 0.814i)7-s + (−0.654 + 0.755i)8-s + (0.913 − 0.406i)9-s + (0.161 + 0.986i)10-s + (−0.710 + 0.703i)12-s + (−0.625 − 0.780i)13-s + (−0.327 + 0.945i)14-s + (0.0855 + 0.996i)15-s + (0.415 − 0.909i)16-s + (0.640 + 0.768i)17-s + (−0.761 + 0.647i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.931 - 0.364i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (576, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.931 - 0.364i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8976239237 - 0.1691986754i\)
\(L(\frac12)\) \(\approx\) \(0.8976239237 - 0.1691986754i\)
\(L(1)\) \(\approx\) \(0.6276453758 - 0.07480284271i\)
\(L(1)\) \(\approx\) \(0.6276453758 - 0.07480284271i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.959 + 0.281i)T \)
3 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + (0.123 - 0.992i)T \)
7 \( 1 + (0.580 - 0.814i)T \)
13 \( 1 + (-0.625 - 0.780i)T \)
17 \( 1 + (0.640 + 0.768i)T \)
19 \( 1 + (0.272 - 0.962i)T \)
23 \( 1 + (0.198 + 0.980i)T \)
29 \( 1 + (0.897 - 0.441i)T \)
37 \( 1 + (0.548 + 0.836i)T \)
41 \( 1 + (0.380 + 0.924i)T \)
43 \( 1 + (0.483 - 0.875i)T \)
47 \( 1 + (-0.564 + 0.825i)T \)
53 \( 1 + (0.749 + 0.662i)T \)
59 \( 1 + (0.380 - 0.924i)T \)
61 \( 1 + (-0.564 + 0.825i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (-0.0665 + 0.997i)T \)
73 \( 1 + (0.580 + 0.814i)T \)
79 \( 1 + (0.820 - 0.572i)T \)
83 \( 1 + (0.749 + 0.662i)T \)
89 \( 1 + (0.696 - 0.717i)T \)
97 \( 1 + (0.516 + 0.856i)T \)
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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

−18.47031078488802856492088253228, −18.07237856436088373162579062623, −17.5894779992453277606410534953, −16.53638182596655079541322238749, −16.34206272802829236329833609954, −15.371861245235112294810434292292, −14.622063024791361455585107128357, −14.01511463385431951719198508842, −12.717915118499542478598614787862, −12.07190763636736089756335453274, −11.711252558866007442049510965224, −11.00399478234728857693171378083, −10.36296956789957225236235335519, −9.75832807153174150800529703778, −9.00046795625158645685203054893, −7.96940548646413630731072780305, −7.398583073477643460598504963790, −6.71747302583317746877376038479, −6.06552732147887491822186295726, −5.298557410778539435695342976443, −4.3113003850028125137207155840, −3.19179063391996047549261914898, −2.30458385889255600835520742302, −1.79442389950008497195676642198, −0.6427935755125521225930352116, 0.88097991046458625421011085533, 0.98989946742091153120981388427, 2.13179689958027703248241035990, 3.43942394233097883780320397221, 4.59080093469707573763118595856, 5.063231197869355287553784941229, 5.79435004766327327910248976355, 6.53802483476993264501632088781, 7.55230587563956079633907313474, 7.84540420347842656475381408537, 8.8010116380334462751145391566, 9.70613509268901853214034959743, 10.07917528199453697354276501655, 10.85077041962669036935251605202, 11.5167442240384877389981815298, 12.13882379058275504020740903603, 12.92442433002735214855135767950, 13.71128675675533896767164767636, 14.76943541694634635200743988206, 15.43765426347061450446402941320, 16.03441691096290347541142173828, 16.80556679914224718909637943429, 17.2889690727320220140688506963, 17.53553036112154554715070769964, 18.241905235053030058929062285563

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