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Properties

Label 1-3751-3751.630-r0-0-0
Degree $1$
Conductor $3751$
Sign $-0.999 + 0.00447i$
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  + (−0.959 − 0.281i)2-s + (0.669 − 0.743i)3-s + (0.841 + 0.540i)4-s + (0.797 − 0.603i)5-s + (−0.851 + 0.524i)6-s + (−0.995 + 0.0950i)7-s + (−0.654 − 0.755i)8-s + (−0.104 − 0.994i)9-s + (−0.935 + 0.353i)10-s + (0.964 − 0.263i)12-s + (0.988 + 0.151i)13-s + (0.981 + 0.189i)14-s + (0.0855 − 0.996i)15-s + (0.415 + 0.909i)16-s + (0.345 + 0.938i)17-s + (−0.179 + 0.983i)18-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.669 − 0.743i)3-s + (0.841 + 0.540i)4-s + (0.797 − 0.603i)5-s + (−0.851 + 0.524i)6-s + (−0.995 + 0.0950i)7-s + (−0.654 − 0.755i)8-s + (−0.104 − 0.994i)9-s + (−0.935 + 0.353i)10-s + (0.964 − 0.263i)12-s + (0.988 + 0.151i)13-s + (0.981 + 0.189i)14-s + (0.0855 − 0.996i)15-s + (0.415 + 0.909i)16-s + (0.345 + 0.938i)17-s + (−0.179 + 0.983i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.999 + 0.00447i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (630, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ -0.999 + 0.00447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002198867632 - 0.9819182943i\)
\(L(\frac12)\) \(\approx\) \(0.002198867632 - 0.9819182943i\)
\(L(1)\) \(\approx\) \(0.7141348907 - 0.4618097882i\)
\(L(1)\) \(\approx\) \(0.7141348907 - 0.4618097882i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.959 - 0.281i)T \)
3 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + (0.797 - 0.603i)T \)
7 \( 1 + (-0.995 + 0.0950i)T \)
13 \( 1 + (0.988 + 0.151i)T \)
17 \( 1 + (0.345 + 0.938i)T \)
19 \( 1 + (-0.969 - 0.244i)T \)
23 \( 1 + (0.198 - 0.980i)T \)
29 \( 1 + (0.897 + 0.441i)T \)
37 \( 1 + (0.449 + 0.893i)T \)
41 \( 1 + (-0.991 + 0.132i)T \)
43 \( 1 + (-0.999 - 0.0190i)T \)
47 \( 1 + (-0.564 - 0.825i)T \)
53 \( 1 + (-0.948 - 0.318i)T \)
59 \( 1 + (-0.991 - 0.132i)T \)
61 \( 1 + (-0.564 - 0.825i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (-0.830 + 0.556i)T \)
73 \( 1 + (-0.995 - 0.0950i)T \)
79 \( 1 + (-0.905 + 0.424i)T \)
83 \( 1 + (-0.948 - 0.318i)T \)
89 \( 1 + (0.696 + 0.717i)T \)
97 \( 1 + (0.516 - 0.856i)T \)
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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

−18.818341817467689475220266832123, −18.4738754451267827058039227005, −17.481575757045600534286612170826, −16.92787304047592777788073111151, −16.07684272744255236822367523919, −15.76813698632236751459992562813, −14.98141150264342595940637882142, −14.30440150247972469821663880974, −13.613432688032611399396794827673, −13.02197123343479737472895875058, −11.74641979557295790283806154006, −10.95316821620106086868442996457, −10.312007764973767938502183637811, −9.88710184146742213076682939291, −9.214073426364973806698320043148, −8.71209645627293505642384076553, −7.77069978526851741190304840850, −7.06649366990820380367546553914, −6.2040386792300314119759621988, −5.771298307547151978254007292574, −4.70195775825757512793783839321, −3.42923698049575064373582861163, −3.01386930735505509610131434238, −2.18576090698214718646768025733, −1.26257029759838319881050835804, 0.33372123725932154043447968527, 1.41377791596512102040225318310, 1.85532132141923652891588869231, 2.877563288504250840695017302800, 3.383996975210560316859039425079, 4.46505096244444092601636380653, 5.93592621186985169225605741880, 6.45926106148246075943760341351, 6.80550593215648785723267932843, 8.10353736845074890698294661720, 8.58802608977428506663919001299, 8.9315870632175027782127698381, 9.93473869679296043789831399768, 10.246109953493401262922082175248, 11.29880576128150176922279978722, 12.284171638224228317681514917432, 12.76158867014766149870882109244, 13.18919883490525318863691138413, 13.97517394484983203249140051755, 14.96487253540752639484762805504, 15.621401839243801148918293922850, 16.54323205727466526048799017916, 16.93919944854707112357859072955, 17.68803771708834768266187896718, 18.53213435072098502923546563560

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