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Properties

Label 1-837-837.265-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.983 - 0.181i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (−0.719 + 0.694i)2-s + (0.0348 − 0.999i)4-s + (−0.939 − 0.342i)5-s + (0.990 − 0.139i)7-s + (0.669 + 0.743i)8-s + (0.913 − 0.406i)10-s + (−0.990 + 0.139i)11-s + (0.241 − 0.970i)13-s + (−0.615 + 0.788i)14-s + (−0.997 − 0.0697i)16-s + (−0.309 + 0.951i)17-s + (−0.104 + 0.994i)19-s + (−0.374 + 0.927i)20-s + (0.615 − 0.788i)22-s + (0.615 − 0.788i)23-s + ⋯
L(s)  = 1  + (−0.719 + 0.694i)2-s + (0.0348 − 0.999i)4-s + (−0.939 − 0.342i)5-s + (0.990 − 0.139i)7-s + (0.669 + 0.743i)8-s + (0.913 − 0.406i)10-s + (−0.990 + 0.139i)11-s + (0.241 − 0.970i)13-s + (−0.615 + 0.788i)14-s + (−0.997 − 0.0697i)16-s + (−0.309 + 0.951i)17-s + (−0.104 + 0.994i)19-s + (−0.374 + 0.927i)20-s + (0.615 − 0.788i)22-s + (0.615 − 0.788i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.983 - 0.181i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.983 - 0.181i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0009485038092 + 0.01034411990i\)
\(L(\frac12)\) \(\approx\) \(0.0009485038092 + 0.01034411990i\)
\(L(1)\) \(\approx\) \(0.5956806464 + 0.09007855107i\)
\(L(1)\) \(\approx\) \(0.5956806464 + 0.09007855107i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.719 + 0.694i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (0.990 - 0.139i)T \)
11 \( 1 + (-0.990 + 0.139i)T \)
13 \( 1 + (0.241 - 0.970i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.615 - 0.788i)T \)
29 \( 1 + (0.719 - 0.694i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.997 + 0.0697i)T \)
43 \( 1 + (-0.961 - 0.275i)T \)
47 \( 1 + (0.559 - 0.829i)T \)
53 \( 1 + (0.978 - 0.207i)T \)
59 \( 1 + (-0.719 - 0.694i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.848 + 0.529i)T \)
83 \( 1 + (-0.961 - 0.275i)T \)
89 \( 1 + (0.978 + 0.207i)T \)
97 \( 1 + (-0.374 + 0.927i)T \)
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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

−21.37879121391209728351513761088, −20.644935121255867339139499633790, −19.89372017213529752843643710517, −19.052394590860381245689197578598, −18.405411468071645868762775196, −17.825653967897943238614218342380, −16.83995186241345711687257515417, −15.85280916689326698559107375308, −15.3809191683753443477538308095, −14.10431133570738760459652636057, −13.32255480583020217990408311803, −12.18796295060522015674879805989, −11.42430772829370666125643355389, −11.09622126693545507009583750778, −10.148545965515303429990735632262, −8.8827997138024337477539731502, −8.45710557225609045478514443691, −7.36287358583993446427095926660, −6.95013345061258535240074577258, −5.08482228142493189954238551424, −4.38387514514779842808459309030, −3.212606388311815333367552999023, −2.41278400123852270478514350768, −1.21293118822764901450924531695, −0.003622643497831929836234549124, 1.01669272556329246030908971773, 2.16795445435151854165724295351, 3.711358115120304134308518373412, 4.84809514944363934355249810305, 5.41665008143607554182253366943, 6.65159109380195332947498306886, 7.67534909393994254357761514374, 8.277900916349836534647294305531, 8.60715496265901796240782945317, 10.26956336948342304875649485267, 10.57821683235305955778981405691, 11.588096196350002500810245912249, 12.591158494995347400352551132716, 13.571208109618322042734050976173, 14.70519259184255007516505459164, 15.25852982904955628949068873261, 15.84002738750686719182586669700, 16.87912592428604379399982541572, 17.434068290840819514176806372549, 18.43957408857713430056517823568, 18.88256563214637164282578729623, 20.05012973793381875461614910823, 20.45614320606256175028755992047, 21.357602015569104819538581629844, 22.87123746740762106975941383225

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