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Properties

Label 1-2675-2675.139-r1-0-0
Degree $1$
Conductor $2675$
Sign $-0.999 + 0.0179i$
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  + (−0.573 + 0.819i)2-s + (−0.815 + 0.578i)3-s + (−0.342 − 0.939i)4-s + (−0.00592 − 0.999i)6-s + (−0.984 − 0.176i)7-s + (0.966 + 0.257i)8-s + (0.331 − 0.943i)9-s + (0.523 − 0.851i)11-s + (0.822 + 0.568i)12-s + (−0.639 + 0.768i)13-s + (0.709 − 0.705i)14-s + (−0.765 + 0.643i)16-s + (−0.112 + 0.993i)17-s + (0.582 + 0.812i)18-s + (0.878 + 0.477i)19-s + ⋯
L(s)  = 1  + (−0.573 + 0.819i)2-s + (−0.815 + 0.578i)3-s + (−0.342 − 0.939i)4-s + (−0.00592 − 0.999i)6-s + (−0.984 − 0.176i)7-s + (0.966 + 0.257i)8-s + (0.331 − 0.943i)9-s + (0.523 − 0.851i)11-s + (0.822 + 0.568i)12-s + (−0.639 + 0.768i)13-s + (0.709 − 0.705i)14-s + (−0.765 + 0.643i)16-s + (−0.112 + 0.993i)17-s + (0.582 + 0.812i)18-s + (0.878 + 0.477i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2675\)    =    \(5^{2} \cdot 107\)
Sign: $-0.999 + 0.0179i$
Analytic conductor: \(287.468\)
Root analytic conductor: \(287.468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2675} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2675,\ (1:\ ),\ -0.999 + 0.0179i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006800623204 + 0.7596302286i\)
\(L(\frac12)\) \(\approx\) \(0.006800623204 + 0.7596302286i\)
\(L(1)\) \(\approx\) \(0.4570210522 + 0.3176755968i\)
\(L(1)\) \(\approx\) \(0.4570210522 + 0.3176755968i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
107 \( 1 \)
good2 \( 1 + (-0.573 + 0.819i)T \)
3 \( 1 + (-0.815 + 0.578i)T \)
7 \( 1 + (-0.984 - 0.176i)T \)
11 \( 1 + (0.523 - 0.851i)T \)
13 \( 1 + (-0.639 + 0.768i)T \)
17 \( 1 + (-0.112 + 0.993i)T \)
19 \( 1 + (0.878 + 0.477i)T \)
23 \( 1 + (0.159 + 0.987i)T \)
29 \( 1 + (0.772 + 0.634i)T \)
31 \( 1 + (-0.461 - 0.886i)T \)
37 \( 1 + (-0.998 + 0.0474i)T \)
41 \( 1 + (0.854 + 0.518i)T \)
43 \( 1 + (-0.205 + 0.978i)T \)
47 \( 1 + (0.229 + 0.973i)T \)
53 \( 1 + (-0.945 - 0.325i)T \)
59 \( 1 + (0.364 + 0.931i)T \)
61 \( 1 + (-0.135 + 0.990i)T \)
67 \( 1 + (0.0533 - 0.998i)T \)
71 \( 1 + (0.297 - 0.954i)T \)
73 \( 1 + (0.217 + 0.976i)T \)
79 \( 1 + (0.171 + 0.985i)T \)
83 \( 1 + (0.553 - 0.832i)T \)
89 \( 1 + (0.952 - 0.303i)T \)
97 \( 1 + (0.657 - 0.753i)T \)
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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

−18.83282614703001524230483340732, −18.03599212248994133556509532052, −17.550669590237705293984388546305, −16.966844176228330973938429504770, −16.074467798161769883288527153350, −15.64568036091500646823421528471, −14.2523624664382517823345561662, −13.49557391690643261171836837991, −12.73640981832202385815210606196, −12.19388679403278274008206946402, −11.85383619041035938475615634330, −10.79363615393005855305330938466, −10.1882942471005836423297735273, −9.55229416843065109173438821915, −8.809597367155080569880625752637, −7.7211043280029657636941631092, −7.05429343765445339925962553719, −6.591088134926220009883331215665, −5.280526316860658785019531729282, −4.74543797117086253890758141548, −3.58776858633699886158461587755, −2.67511316758772601080416849992, −2.03838783399632055669116727372, −0.78779562751734868525822815972, −0.320863925611239894118846095089, 0.74859006998734235681450346525, 1.572199828629298045173606488990, 3.202818732417136755587514445447, 3.98268359999346614487410631443, 4.8146685800661895205511924293, 5.80084514839664657215480557631, 6.17215928989685097677214732602, 6.91516041961483885212745358850, 7.68102997803222008581686950865, 8.811872878104329995421068145944, 9.43833983270846975670227603605, 9.89392743860048204782271843483, 10.73088451987835736463374836214, 11.40013533504406619648322217127, 12.2525508531047408325689486525, 13.14662262477914798251668290406, 14.044835970277241964459789347066, 14.66373954496631872661700952088, 15.55786624527225458402344861549, 16.12222081079957183214698454621, 16.6573116615728092259475594049, 17.12078617675409722222697381533, 17.8707591427209450724514710524, 18.68535768388814493773572391074, 19.4505296057374368270466411151

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