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Properties

Label 1-3751-3751.1344-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.652 - 0.757i$
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  + (−0.415 + 0.909i)2-s + (−0.913 − 0.406i)3-s + (−0.654 − 0.755i)4-s + (−0.179 + 0.983i)5-s + (0.749 − 0.662i)6-s + (0.786 − 0.618i)7-s + (0.959 − 0.281i)8-s + (0.669 + 0.743i)9-s + (−0.820 − 0.572i)10-s + (0.290 + 0.956i)12-s + (0.483 + 0.875i)13-s + (0.235 + 0.971i)14-s + (0.564 − 0.825i)15-s + (−0.142 + 0.989i)16-s + (−0.625 + 0.780i)17-s + (−0.953 + 0.299i)18-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (−0.913 − 0.406i)3-s + (−0.654 − 0.755i)4-s + (−0.179 + 0.983i)5-s + (0.749 − 0.662i)6-s + (0.786 − 0.618i)7-s + (0.959 − 0.281i)8-s + (0.669 + 0.743i)9-s + (−0.820 − 0.572i)10-s + (0.290 + 0.956i)12-s + (0.483 + 0.875i)13-s + (0.235 + 0.971i)14-s + (0.564 − 0.825i)15-s + (−0.142 + 0.989i)16-s + (−0.625 + 0.780i)17-s + (−0.953 + 0.299i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.652 - 0.757i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (1344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.652 - 0.757i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3522903547 - 0.1614548043i\)
\(L(\frac12)\) \(\approx\) \(0.3522903547 - 0.1614548043i\)
\(L(1)\) \(\approx\) \(0.5421360693 + 0.2143159921i\)
\(L(1)\) \(\approx\) \(0.5421360693 + 0.2143159921i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.415 + 0.909i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (-0.179 + 0.983i)T \)
7 \( 1 + (0.786 - 0.618i)T \)
13 \( 1 + (0.483 + 0.875i)T \)
17 \( 1 + (-0.625 + 0.780i)T \)
19 \( 1 + (-0.161 + 0.986i)T \)
23 \( 1 + (0.985 - 0.170i)T \)
29 \( 1 + (-0.998 - 0.0570i)T \)
37 \( 1 + (-0.123 - 0.992i)T \)
41 \( 1 + (0.595 - 0.803i)T \)
43 \( 1 + (-0.991 - 0.132i)T \)
47 \( 1 + (-0.870 - 0.491i)T \)
53 \( 1 + (-0.640 + 0.768i)T \)
59 \( 1 + (-0.595 - 0.803i)T \)
61 \( 1 + (-0.870 - 0.491i)T \)
67 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (0.548 - 0.836i)T \)
73 \( 1 + (-0.786 - 0.618i)T \)
79 \( 1 + (0.997 + 0.0760i)T \)
83 \( 1 + (0.640 - 0.768i)T \)
89 \( 1 + (-0.774 + 0.633i)T \)
97 \( 1 + (0.610 - 0.791i)T \)
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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

−18.40745124832565074996544529412, −18.1247808044952101846860711170, −17.32665527339178214070231884810, −16.97135171746866463714969138424, −16.06494828331611793683120330424, −15.493505752039383997437179186639, −14.766656516695340903906296347, −13.35145467537726366774769285708, −13.10692389703162820695254005167, −12.30227823686327671339810872705, −11.55171012365113742946485610661, −11.253048291291972130367044604196, −10.6000231434456027011595417932, −9.47431444884263758736921270413, −9.2086831343699664396545359390, −8.37199487778905366659348935185, −7.720603476078628542109491397224, −6.67140009585671387116717440787, −5.52250768651132317457572939247, −4.90962410785751891410056593892, −4.59545357297129019199285540768, −3.544473553749011388873764933663, −2.654789840026964365967500410456, −1.50509361678355271000383462951, −0.92185948564973802397612234971, 0.18163802480833328752296961405, 1.53965618192201944963069303334, 1.92603023425124644944065560980, 3.698569064578945998253427015614, 4.28913894305346013590783120097, 5.13178941994808689077757551470, 6.00523691412586110244551580061, 6.53714660562743096400500448846, 7.16918861201273819169844457913, 7.73000564017325352178846525071, 8.44145705945667321628040711214, 9.435446823067973041267825342702, 10.363037783131276767817185634999, 10.96121852792225680205464407817, 11.17550950136106239239043797146, 12.28306900794094274018983590895, 13.22311224673380175044023215036, 13.8561229711605647488296094548, 14.51099413367520213904065314014, 15.11693004230519336315079987612, 15.8780023106828002107160176064, 16.77392686467833066360675968027, 17.01014162311985033359426219102, 17.904283695285751803123469203938, 18.249086171501169886404438183673

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