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Properties

Label 1-2675-2675.2504-r1-0-0
Degree $1$
Conductor $2675$
Sign $0.623 - 0.781i$
Analytic cond. $287.468$
Root an. cond. $287.468$
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  + (−0.553 − 0.832i)2-s + (−0.523 − 0.851i)3-s + (−0.386 + 0.922i)4-s + (−0.419 + 0.907i)6-s + (−0.915 + 0.403i)7-s + (0.982 − 0.188i)8-s + (−0.451 + 0.892i)9-s + (0.563 − 0.826i)11-s + (0.988 − 0.153i)12-s + (−0.353 − 0.935i)13-s + (0.842 + 0.538i)14-s + (−0.700 − 0.713i)16-s + (−0.933 + 0.359i)17-s + (0.992 − 0.118i)18-s + (0.217 + 0.976i)19-s + ⋯
L(s)  = 1  + (−0.553 − 0.832i)2-s + (−0.523 − 0.851i)3-s + (−0.386 + 0.922i)4-s + (−0.419 + 0.907i)6-s + (−0.915 + 0.403i)7-s + (0.982 − 0.188i)8-s + (−0.451 + 0.892i)9-s + (0.563 − 0.826i)11-s + (0.988 − 0.153i)12-s + (−0.353 − 0.935i)13-s + (0.842 + 0.538i)14-s + (−0.700 − 0.713i)16-s + (−0.933 + 0.359i)17-s + (0.992 − 0.118i)18-s + (0.217 + 0.976i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2675\)    =    \(5^{2} \cdot 107\)
Sign: $0.623 - 0.781i$
Analytic conductor: \(287.468\)
Root analytic conductor: \(287.468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2675} (2504, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2675,\ (1:\ ),\ 0.623 - 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4267894595 - 0.2054848712i\)
\(L(\frac12)\) \(\approx\) \(0.4267894595 - 0.2054848712i\)
\(L(1)\) \(\approx\) \(0.4269796785 - 0.2927996891i\)
\(L(1)\) \(\approx\) \(0.4269796785 - 0.2927996891i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
107 \( 1 \)
good2 \( 1 + (-0.553 - 0.832i)T \)
3 \( 1 + (-0.523 - 0.851i)T \)
7 \( 1 + (-0.915 + 0.403i)T \)
11 \( 1 + (0.563 - 0.826i)T \)
13 \( 1 + (-0.353 - 0.935i)T \)
17 \( 1 + (-0.933 + 0.359i)T \)
19 \( 1 + (0.217 + 0.976i)T \)
23 \( 1 + (-0.772 - 0.634i)T \)
29 \( 1 + (0.997 + 0.0710i)T \)
31 \( 1 + (0.472 - 0.881i)T \)
37 \( 1 + (0.949 + 0.314i)T \)
41 \( 1 + (-0.513 - 0.858i)T \)
43 \( 1 + (-0.533 + 0.845i)T \)
47 \( 1 + (-0.919 + 0.392i)T \)
53 \( 1 + (-0.620 - 0.783i)T \)
59 \( 1 + (0.848 + 0.528i)T \)
61 \( 1 + (0.503 + 0.864i)T \)
67 \( 1 + (-0.683 - 0.729i)T \)
71 \( 1 + (-0.0769 - 0.997i)T \)
73 \( 1 + (-0.297 + 0.954i)T \)
79 \( 1 + (-0.0177 - 0.999i)T \)
83 \( 1 + (-0.910 + 0.413i)T \)
89 \( 1 + (-0.872 - 0.487i)T \)
97 \( 1 + (0.867 - 0.498i)T \)
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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

−19.34359399733334801289206804550, −18.187033832616679313266536523182, −17.62350935272501605606306041320, −17.074760312271598778547938473937, −16.36720470885480249517288583371, −15.79006070561424633675456948745, −15.34120832163343093219070049859, −14.40968474450328914273006745082, −13.822912205287980696908371546006, −12.91551908401201607203283825634, −11.82354364126824128905102832796, −11.27230717360322166771115521193, −10.212359887255533601147440263528, −9.78308425083398135933482578576, −9.24131153913403007724373260472, −8.549799411682723107886240855907, −7.26679372494661255243886834735, −6.67026940014819943291981677535, −6.30571004616402026266728520669, −5.09816938916196348676691795152, −4.54118351728158366670019491616, −3.8802316471381594194773485259, −2.62292462982480686114017439753, −1.351941747264584257317277402238, −0.2332351421002541607287586750, 0.42964944174438042222393334399, 1.30007007660324971257150880733, 2.32482123615361360883001100622, 2.95326938531653997043068084212, 3.854284110285990896095952166715, 4.908130570401207982384748362518, 6.08236281207429612733854465920, 6.39176094851925996410615305403, 7.4975737240118686649319342603, 8.27948675083259607343308685557, 8.734237646607416014975881689395, 9.91474642745277669698475582941, 10.30031795399715139664122765533, 11.31344777700270320637470896759, 11.82913380910544661892505418184, 12.53386697901714354440899644870, 13.07014342040723657162392934650, 13.673096652225812648749123544024, 14.60929478950147317875163812317, 15.848381929160472750566187440195, 16.45760949258597768710043206794, 17.05444903356206755907797216478, 17.86316465736268987755224149294, 18.350100215083095304410209090145, 19.08512440379817810322584711911

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