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Properties

Label 1-3751-3751.1028-r0-0-0
Degree $1$
Conductor $3751$
Sign $-0.281 + 0.959i$
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 + (0.669 − 0.743i)3-s + (0.993 − 0.113i)4-s + (−0.532 − 0.846i)5-s + (−0.625 + 0.780i)6-s + (0.797 + 0.603i)7-s + (−0.985 + 0.170i)8-s + (−0.104 − 0.994i)9-s + (0.580 + 0.814i)10-s + (0.580 − 0.814i)12-s + (0.861 − 0.508i)13-s + (−0.830 − 0.556i)14-s + (−0.985 − 0.170i)15-s + (0.974 − 0.226i)16-s + (−0.179 − 0.983i)17-s + (0.161 + 0.986i)18-s + ⋯
L(s)  = 1  + (−0.998 + 0.0570i)2-s + (0.669 − 0.743i)3-s + (0.993 − 0.113i)4-s + (−0.532 − 0.846i)5-s + (−0.625 + 0.780i)6-s + (0.797 + 0.603i)7-s + (−0.985 + 0.170i)8-s + (−0.104 − 0.994i)9-s + (0.580 + 0.814i)10-s + (0.580 − 0.814i)12-s + (0.861 − 0.508i)13-s + (−0.830 − 0.556i)14-s + (−0.985 − 0.170i)15-s + (0.974 − 0.226i)16-s + (−0.179 − 0.983i)17-s + (0.161 + 0.986i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.281 + 0.959i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (1028, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ -0.281 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07005001637 + 0.09353488884i\)
\(L(\frac12)\) \(\approx\) \(0.07005001637 + 0.09353488884i\)
\(L(1)\) \(\approx\) \(0.6712905259 - 0.2226962834i\)
\(L(1)\) \(\approx\) \(0.6712905259 - 0.2226962834i\)

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 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + (-0.532 - 0.846i)T \)
7 \( 1 + (0.797 + 0.603i)T \)
13 \( 1 + (0.861 - 0.508i)T \)
17 \( 1 + (-0.179 - 0.983i)T \)
19 \( 1 + (-0.851 + 0.524i)T \)
23 \( 1 + (-0.654 + 0.755i)T \)
29 \( 1 + (-0.564 + 0.825i)T \)
37 \( 1 + (-0.217 + 0.976i)T \)
41 \( 1 + (-0.710 - 0.703i)T \)
43 \( 1 + (0.928 - 0.371i)T \)
47 \( 1 + (0.774 + 0.633i)T \)
53 \( 1 + (-0.683 + 0.730i)T \)
59 \( 1 + (-0.710 + 0.703i)T \)
61 \( 1 + (-0.998 - 0.0570i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (-0.991 + 0.132i)T \)
73 \( 1 + (-0.999 + 0.0190i)T \)
79 \( 1 + (0.272 - 0.962i)T \)
83 \( 1 + (-0.905 - 0.424i)T \)
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

−18.469829717527478304514626146467, −17.80069763511334278364622845342, −16.989727798425354267863463460370, −16.43933470361715482346585732650, −15.58274216223979255076258327721, −15.159828683321600245176962769399, −14.50078644247880198575853734874, −13.90197077845288737546950006624, −12.905838479337819558859665404934, −11.79301558702568916359645866994, −11.04001176725523230874670390351, −10.762556041729191043019456312824, −10.19504571780215139162436110831, −9.27064487599258537904432725881, −8.46629186735610414045984900146, −8.1056236321134650697850502551, −7.38390654085083193881438899715, −6.578360717041008280589635737613, −5.81831407965771519911175067770, −4.31638535914886444411527954489, −4.0422731239924094338628433316, −3.11312285088985325197997616985, −2.21049822466547022454968267401, −1.60442586994669073968220774168, −0.03975356807811213452726326710, 1.29515580079869830553343397669, 1.56276680993127280520969686007, 2.61053383999298467370705995664, 3.408895467371142367005120203122, 4.4143965443808680314230943979, 5.588457695999409526411432403494, 6.07700596409734247111778566615, 7.28040164888186674851085600794, 7.67885967272923146552873493002, 8.39997431279280964270382962611, 8.857142929419312450325465522412, 9.316848751347365225969600547023, 10.46282192561884669720192521074, 11.26126339993254992869769301645, 11.99324186326071425090398142008, 12.332610408662601448814885370756, 13.26061632571008929091452542498, 14.047780465067060010088352021063, 14.91753346829695250461370316167, 15.52015360967844670389771377687, 15.97967434023487871205250095605, 16.97958102360456576710759810287, 17.571784965960613853428924872713, 18.26198453344090089772114256216, 18.74826913214932376269770599719

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