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Properties

Label 1-3751-3751.1961-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.552 + 0.833i$
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  + (0.941 − 0.336i)2-s + (0.309 − 0.951i)3-s + (0.774 − 0.633i)4-s + (−0.921 − 0.389i)5-s + (−0.0285 − 0.999i)6-s + (−0.736 − 0.676i)7-s + (0.516 − 0.856i)8-s + (−0.809 − 0.587i)9-s + (−0.998 − 0.0570i)10-s + (−0.362 − 0.931i)12-s + (−0.254 + 0.967i)13-s + (−0.921 − 0.389i)14-s + (−0.654 + 0.755i)15-s + (0.198 − 0.980i)16-s + (−0.142 − 0.989i)17-s + (−0.959 − 0.281i)18-s + ⋯
L(s)  = 1  + (0.941 − 0.336i)2-s + (0.309 − 0.951i)3-s + (0.774 − 0.633i)4-s + (−0.921 − 0.389i)5-s + (−0.0285 − 0.999i)6-s + (−0.736 − 0.676i)7-s + (0.516 − 0.856i)8-s + (−0.809 − 0.587i)9-s + (−0.998 − 0.0570i)10-s + (−0.362 − 0.931i)12-s + (−0.254 + 0.967i)13-s + (−0.921 − 0.389i)14-s + (−0.654 + 0.755i)15-s + (0.198 − 0.980i)16-s + (−0.142 − 0.989i)17-s + (−0.959 − 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.552 + 0.833i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (1961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.552 + 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05396539283 + 0.02897213595i\)
\(L(\frac12)\) \(\approx\) \(-0.05396539283 + 0.02897213595i\)
\(L(1)\) \(\approx\) \(0.9492055630 - 0.8080680441i\)
\(L(1)\) \(\approx\) \(0.9492055630 - 0.8080680441i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.941 - 0.336i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.921 - 0.389i)T \)
7 \( 1 + (-0.736 - 0.676i)T \)
13 \( 1 + (-0.254 + 0.967i)T \)
17 \( 1 + (-0.142 - 0.989i)T \)
19 \( 1 + (-0.142 + 0.989i)T \)
23 \( 1 + (0.993 - 0.113i)T \)
29 \( 1 + (-0.985 + 0.170i)T \)
37 \( 1 + (-0.998 - 0.0570i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (0.0855 + 0.996i)T \)
47 \( 1 + (-0.0285 + 0.999i)T \)
53 \( 1 + (-0.870 + 0.491i)T \)
59 \( 1 + (-0.959 + 0.281i)T \)
61 \( 1 + (-0.564 - 0.825i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (0.993 - 0.113i)T \)
79 \( 1 + (0.974 - 0.226i)T \)
83 \( 1 + (-0.736 - 0.676i)T \)
89 \( 1 + (0.897 - 0.441i)T \)
97 \( 1 + (-0.654 - 0.755i)T \)
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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

−19.47681746518374492109580380522, −18.72834817142587540779742819389, −17.51163655082509557334533305420, −16.86958043139013191002379618642, −16.17211450521857583428205265954, −15.35617050366076679879268841529, −15.20884650557551065047328099604, −14.84853450625189989678351908130, −13.690523552032078278367025305996, −13.08881245839196668194932177399, −12.329147036318017990480748925125, −11.69031697106283317865831219973, −10.799025968379443641496737349351, −10.50475266195304387406865787440, −9.319229331008077741004040116201, −8.58278725729675315265591437693, −7.97101912381027944391306173856, −7.08987807323773215784389747930, −6.38612934619517523353417907132, −5.43455614477108987566115728977, −4.961933146696318006632789874146, −3.9997153643806770813635780953, −3.36085906013914963655370551343, −2.95799175955589522552344172246, −2.077786902796739951169029759957, 0.01146546764135232575786863558, 1.134913368174718594542450222133, 1.83812276710155539243793221130, 3.01600203738193762260297140068, 3.406787839097754272728785445177, 4.289602065619238630102811144503, 4.982610141779426427714573810726, 6.03203194793329898855326500858, 6.77857084415517636768877789709, 7.27283503724005359641107049866, 7.85280269693790134308186869714, 9.034888805234309641942007210402, 9.55405735529473603281197579544, 10.72871622902637037499648906521, 11.34733148804029560811463646623, 12.04834843100994716681025027077, 12.552950756775558220770052457618, 13.12472098086720607844179405075, 13.86761252003986471609693414213, 14.34303065106268353168338967243, 15.14349019505041250600643055869, 15.91568409492874262789740066525, 16.62426425392086645054705826295, 17.07178505775165344632942324976, 18.4758865136110498814237090644

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