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Properties

Label 1-3751-3751.999-r0-0-0
Degree $1$
Conductor $3751$
Sign $-0.305 - 0.952i$
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 + (−0.104 − 0.994i)3-s + (−0.654 + 0.755i)4-s + (−0.991 − 0.132i)5-s + (0.861 − 0.508i)6-s + (−0.786 − 0.618i)7-s + (−0.959 − 0.281i)8-s + (−0.978 + 0.207i)9-s + (−0.290 − 0.956i)10-s + (0.820 + 0.572i)12-s + (0.123 + 0.992i)13-s + (0.235 − 0.971i)14-s + (−0.0285 + 0.999i)15-s + (−0.142 − 0.989i)16-s + (0.548 − 0.836i)17-s + (−0.595 − 0.803i)18-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (−0.104 − 0.994i)3-s + (−0.654 + 0.755i)4-s + (−0.991 − 0.132i)5-s + (0.861 − 0.508i)6-s + (−0.786 − 0.618i)7-s + (−0.959 − 0.281i)8-s + (−0.978 + 0.207i)9-s + (−0.290 − 0.956i)10-s + (0.820 + 0.572i)12-s + (0.123 + 0.992i)13-s + (0.235 − 0.971i)14-s + (−0.0285 + 0.999i)15-s + (−0.142 − 0.989i)16-s + (0.548 − 0.836i)17-s + (−0.595 − 0.803i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.305 - 0.952i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ -0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3578306205 - 0.4904704563i\)
\(L(\frac12)\) \(\approx\) \(0.3578306205 - 0.4904704563i\)
\(L(1)\) \(\approx\) \(0.7792644627 + 0.03283922533i\)
\(L(1)\) \(\approx\) \(0.7792644627 + 0.03283922533i\)

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 \( 1 + (-0.104 - 0.994i)T \)
5 \( 1 + (-0.991 - 0.132i)T \)
7 \( 1 + (-0.786 - 0.618i)T \)
13 \( 1 + (0.123 + 0.992i)T \)
17 \( 1 + (0.548 - 0.836i)T \)
19 \( 1 + (-0.710 - 0.703i)T \)
23 \( 1 + (0.897 - 0.441i)T \)
29 \( 1 + (-0.362 - 0.931i)T \)
37 \( 1 + (0.483 + 0.875i)T \)
41 \( 1 + (0.953 + 0.299i)T \)
43 \( 1 + (-0.179 + 0.983i)T \)
47 \( 1 + (0.198 + 0.980i)T \)
53 \( 1 + (-0.0665 - 0.997i)T \)
59 \( 1 + (0.953 - 0.299i)T \)
61 \( 1 + (0.198 + 0.980i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (-0.625 + 0.780i)T \)
73 \( 1 + (-0.786 + 0.618i)T \)
79 \( 1 + (-0.851 - 0.524i)T \)
83 \( 1 + (-0.0665 - 0.997i)T \)
89 \( 1 + (-0.254 - 0.967i)T \)
97 \( 1 + (0.941 - 0.336i)T \)
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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

−19.063494554248584422357115396695, −18.4082344496200657682707317693, −17.46391511595725176938903286490, −16.622565660548976906003090429057, −15.91485155984853776500425461538, −15.09349554063200854094221844355, −15.005398655523450815625608281818, −14.15111010263815212060509786023, −12.92904231176847040173823554578, −12.59291734174600340839462793368, −11.90229737034432525460203938822, −11.10947948656361227380652810136, −10.52843934997319642930900945787, −10.07997869398090257961884642294, −9.0550796880626040838173563697, −8.68321510407736939865349654454, −7.76032208145519307380520333417, −6.52646906897038849641153684541, −5.61835807838711955064066723129, −5.288390808528722842079414891130, −4.11849497102364309853456336581, −3.66796311822250834010044028373, −3.12269781863967614288441132129, −2.32917807498108148236395529438, −0.90354150397736150727626834661, 0.22055108930458905185221968519, 1.06106428682690202442577134727, 2.649172654862530120628523254236, 3.20530541360853771560549354945, 4.25828149329111774977793407358, 4.69372552197081165019868584343, 5.84223091109417177997083405994, 6.56541908951722325586631138928, 7.08201013394147012114362066106, 7.55446663159110214710442126479, 8.36672552236925239444440849926, 9.010535559394184869946752115828, 9.82120871722970919660560738968, 11.24497528143302610339748502351, 11.54320908421264500925909325074, 12.46550067646677687473000600412, 13.03121767290389738923971694669, 13.462820171962982574479364926842, 14.380669175869948562284018749657, 14.81895799301987785756725769739, 15.8737640113475496676325227642, 16.33250599258348148169560750099, 16.920335083774436485655289488299, 17.50732273752741052694244810209, 18.5102752511265181542390835846

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