↑

Properties

Label 1-3751-3751.598-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.515 + 0.856i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.516 + 0.856i)2-s + (−0.5 − 0.866i)3-s + (−0.466 + 0.884i)4-s + (0.964 + 0.263i)5-s + (0.483 − 0.875i)6-s + (0.380 − 0.924i)7-s + (−0.998 + 0.0570i)8-s + (−0.5 + 0.866i)9-s + (0.272 + 0.962i)10-s + (0.999 − 0.0380i)12-s + (−0.532 − 0.846i)13-s + (0.988 − 0.151i)14-s + (−0.254 − 0.967i)15-s + (−0.564 − 0.825i)16-s + (0.861 + 0.508i)17-s + (−0.999 + 0.0190i)18-s + ⋯
L(s)  = 1  + (0.516 + 0.856i)2-s + (−0.5 − 0.866i)3-s + (−0.466 + 0.884i)4-s + (0.964 + 0.263i)5-s + (0.483 − 0.875i)6-s + (0.380 − 0.924i)7-s + (−0.998 + 0.0570i)8-s + (−0.5 + 0.866i)9-s + (0.272 + 0.962i)10-s + (0.999 − 0.0380i)12-s + (−0.532 − 0.846i)13-s + (0.988 − 0.151i)14-s + (−0.254 − 0.967i)15-s + (−0.564 − 0.825i)16-s + (0.861 + 0.508i)17-s + (−0.999 + 0.0190i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.515 + 0.856i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (598, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.515 + 0.856i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.935721224 + 1.094736121i\)
\(L(\frac12)\) \(\approx\) \(1.935721224 + 1.094736121i\)
\(L(1)\) \(\approx\) \(1.311208386 + 0.3815786054i\)
\(L(1)\) \(\approx\) \(1.311208386 + 0.3815786054i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.516 + 0.856i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.964 + 0.263i)T \)
7 \( 1 + (0.380 - 0.924i)T \)
13 \( 1 + (-0.532 - 0.846i)T \)
17 \( 1 + (0.861 + 0.508i)T \)
19 \( 1 + (0.953 + 0.299i)T \)
23 \( 1 + (-0.0285 + 0.999i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (-0.532 + 0.846i)T \)
41 \( 1 + (0.797 - 0.603i)T \)
43 \( 1 + (-0.625 + 0.780i)T \)
47 \( 1 + (-0.654 - 0.755i)T \)
53 \( 1 + (0.723 - 0.690i)T \)
59 \( 1 + (0.797 + 0.603i)T \)
61 \( 1 + (-0.921 + 0.389i)T \)
67 \( 1 + (-0.327 - 0.945i)T \)
71 \( 1 + (0.749 + 0.662i)T \)
73 \( 1 + (-0.761 + 0.647i)T \)
79 \( 1 + (-0.888 - 0.458i)T \)
83 \( 1 + (-0.179 + 0.983i)T \)
89 \( 1 + (0.993 + 0.113i)T \)
97 \( 1 + (-0.998 + 0.0570i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.46155393162807630394566536971, −17.89254826139107824759580528358, −17.25186423589020429107562299096, −16.359919411464782599857052598652, −15.7868616764588882081461686237, −14.80140097157898813882537486869, −14.38356807369731919824410311503, −13.80551541167997201370647667337, −12.80718064143224765732984827100, −11.98205407242310808443065360710, −11.82780830656683202104217720185, −10.87677507684585580120291630207, −10.17357002919049085094746511597, −9.44009510545607325217598926501, −9.250801177992215019780182714434, −8.32714189502466008889615601749, −6.86741607142525308042157307159, −5.9614946043805960364779836497, −5.53658113234101024389264992754, −4.81749414866575155368793651956, −4.37536743522495645956008576954, −3.15368439577357406594212439426, −2.55725207730436841530804529580, −1.72276208593910150941412679960, −0.680223850660442714800605487226, 0.93587395444488894973936055509, 1.70459761031861155092735635049, 2.90288518308723141320424551994, 3.51527273690332136056006464785, 4.80525716654996845215318622024, 5.40003509220989229209791416624, 5.833680843530877211608239378087, 6.78916276697545120766564233766, 7.27420921071335255481765113608, 7.869099311640590211228832913331, 8.58729242257342567379906149624, 9.81301809031762477599728505829, 10.287590017694772199629282580317, 11.262728628640374214347031807348, 12.03879770150823595814009303051, 12.75727661341661520165798849169, 13.35524849254363991744280009975, 13.90477440172177094064931305699, 14.42143436274380832099391389596, 15.10625765181931836087059128565, 16.24978708337327235108847937442, 16.789859918095270081447233249146, 17.326968409865419173646284942929, 17.95113457797056864866646928250, 18.20222620302336640498266327778

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