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Properties

Label 1-3751-3751.1029-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.596 + 0.802i$
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  + (0.998 − 0.0570i)2-s + (0.978 + 0.207i)3-s + (0.993 − 0.113i)4-s + (0.999 − 0.0380i)5-s + (0.988 + 0.151i)6-s + (−0.123 + 0.992i)7-s + (0.985 − 0.170i)8-s + (0.913 + 0.406i)9-s + (0.995 − 0.0950i)10-s + (0.995 + 0.0950i)12-s + (0.00951 + 0.999i)13-s + (−0.0665 + 0.997i)14-s + (0.985 + 0.170i)15-s + (0.974 − 0.226i)16-s + (−0.761 + 0.647i)17-s + (0.935 + 0.353i)18-s + ⋯
L(s)  = 1  + (0.998 − 0.0570i)2-s + (0.978 + 0.207i)3-s + (0.993 − 0.113i)4-s + (0.999 − 0.0380i)5-s + (0.988 + 0.151i)6-s + (−0.123 + 0.992i)7-s + (0.985 − 0.170i)8-s + (0.913 + 0.406i)9-s + (0.995 − 0.0950i)10-s + (0.995 + 0.0950i)12-s + (0.00951 + 0.999i)13-s + (−0.0665 + 0.997i)14-s + (0.985 + 0.170i)15-s + (0.974 − 0.226i)16-s + (−0.761 + 0.647i)17-s + (0.935 + 0.353i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.596 + 0.802i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (1029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.596 + 0.802i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.686497384 + 2.857324517i\)
\(L(\frac12)\) \(\approx\) \(5.686497384 + 2.857324517i\)
\(L(1)\) \(\approx\) \(3.112451444 + 0.7042887045i\)
\(L(1)\) \(\approx\) \(3.112451444 + 0.7042887045i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.998 - 0.0570i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (0.999 - 0.0380i)T \)
7 \( 1 + (-0.123 + 0.992i)T \)
13 \( 1 + (0.00951 + 0.999i)T \)
17 \( 1 + (-0.761 + 0.647i)T \)
19 \( 1 + (-0.879 - 0.475i)T \)
23 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (-0.564 + 0.825i)T \)
37 \( 1 + (-0.953 + 0.299i)T \)
41 \( 1 + (-0.964 + 0.263i)T \)
43 \( 1 + (-0.786 - 0.618i)T \)
47 \( 1 + (0.774 + 0.633i)T \)
53 \( 1 + (0.290 + 0.956i)T \)
59 \( 1 + (0.964 + 0.263i)T \)
61 \( 1 + (-0.998 - 0.0570i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (0.380 - 0.924i)T \)
73 \( 1 + (0.483 - 0.875i)T \)
79 \( 1 + (-0.969 + 0.244i)T \)
83 \( 1 + (0.820 - 0.572i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (-0.466 - 0.884i)T \)
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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

−18.68090312419826466443608967292, −17.60727039268418880322354634288, −17.203938276552088780104640171853, −16.3180894455937357649055331251, −15.49285513222079437597390729783, −14.90849631921980372236449747160, −14.27892300306757204681138115654, −13.540350318652700985730639323580, −13.25331338268996611436699840079, −12.79678649854203792793166809133, −11.74734492810066713905693456706, −10.74857703748650266858184105716, −10.24497531781848810848690860706, −9.57714680240201642359875960268, −8.57517880502124251466203232835, −7.82556392572026095211535834904, −6.96640705724848894990437843441, −6.6521191551715482482324461439, −5.59612832206298151661544986215, −4.90693996968572432777427626727, −3.94280243220029187775896790486, −3.39659639753743764521206734277, −2.52248121028847720477160069907, −1.911022686816287176087028759985, −1.00170789906628740483342537906, 1.70982696951000440135878453420, 1.947818697705832021644431869847, 2.73881911048831565376428696193, 3.436159313307773655616517672487, 4.499571965588461965111370914022, 4.92042847464885924833420270964, 5.91102643153180919309159813827, 6.61947245853107633436161247820, 7.10173797871989005460730612852, 8.47593902188769206502612490700, 8.85618541318602280984111305904, 9.56404490159522696062920800740, 10.52623086462511914939674398891, 10.96831982279356975631596714400, 12.15962213573494969647184773174, 12.65918383325653026085154663850, 13.40245597307372486577807712105, 13.79246318270801034977761497320, 14.68210891646663898483098845467, 15.03529877187855834008294783407, 15.64921489952544606255637250570, 16.57774466198058634728416332038, 17.07337610334753543809679500029, 18.26617909182841969990645155992, 18.87673872169613303574338478018

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