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Properties

Label 1-837-837.704-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.977 + 0.210i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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  + (−0.559 − 0.829i)2-s + (−0.374 + 0.927i)4-s + (−0.766 − 0.642i)5-s + (0.0348 − 0.999i)7-s + (0.978 − 0.207i)8-s + (−0.104 + 0.994i)10-s + (0.0348 − 0.999i)11-s + (−0.438 − 0.898i)13-s + (−0.848 + 0.529i)14-s + (−0.719 − 0.694i)16-s + (0.309 − 0.951i)17-s + (0.913 − 0.406i)19-s + (0.882 − 0.469i)20-s + (−0.848 + 0.529i)22-s + (0.848 − 0.529i)23-s + ⋯
L(s)  = 1  + (−0.559 − 0.829i)2-s + (−0.374 + 0.927i)4-s + (−0.766 − 0.642i)5-s + (0.0348 − 0.999i)7-s + (0.978 − 0.207i)8-s + (−0.104 + 0.994i)10-s + (0.0348 − 0.999i)11-s + (−0.438 − 0.898i)13-s + (−0.848 + 0.529i)14-s + (−0.719 − 0.694i)16-s + (0.309 − 0.951i)17-s + (0.913 − 0.406i)19-s + (0.882 − 0.469i)20-s + (−0.848 + 0.529i)22-s + (0.848 − 0.529i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.977 + 0.210i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (704, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.977 + 0.210i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08308848193 - 0.7804424211i\)
\(L(\frac12)\) \(\approx\) \(-0.08308848193 - 0.7804424211i\)
\(L(1)\) \(\approx\) \(0.4859096038 - 0.5096021366i\)
\(L(1)\) \(\approx\) \(0.4859096038 - 0.5096021366i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.559 - 0.829i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (0.0348 - 0.999i)T \)
11 \( 1 + (0.0348 - 0.999i)T \)
13 \( 1 + (-0.438 - 0.898i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.848 - 0.529i)T \)
29 \( 1 + (0.559 + 0.829i)T \)
37 \( 1 - T \)
41 \( 1 + (0.719 - 0.694i)T \)
43 \( 1 + (0.997 - 0.0697i)T \)
47 \( 1 + (0.241 - 0.970i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (-0.559 + 0.829i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.990 - 0.139i)T \)
83 \( 1 + (-0.997 + 0.0697i)T \)
89 \( 1 + (0.669 - 0.743i)T \)
97 \( 1 + (-0.882 + 0.469i)T \)
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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

−22.80771101961996958330995345055, −21.99374621910213120956077303589, −20.99602381645164645511867593416, −19.7575237038474423241536501630, −19.165382737708259782985101814363, −18.62442663081646714514543543725, −17.74365018916116355034778956882, −17.05119580148567151860273067319, −15.8787549449313774027137944672, −15.49473156972066274194713077291, −14.64623236208976339294184816310, −14.182533546112228105146847587694, −12.731562809291964300679216667238, −11.8960720344975948345375454077, −11.07912807841295446679850516995, −9.9721715176175609708631384995, −9.3487478311657113033899060808, −8.33645006723326957876550865610, −7.56226783858771504399254971388, −6.842376736622048605141891788276, −5.947112877807200245956954044319, −4.93498016555557003242494333782, −4.003695939113970518834308822197, −2.6110222657045209704744435799, −1.49641976288389193502401886631, 0.54572529393142306669609509087, 1.090447312117335632362092905389, 2.8737873063314357770812736938, 3.47131969557945021260084869557, 4.535083237943530362512249401880, 5.33166847274054646446762451206, 7.168372602640147406867846115704, 7.58283922669539582378635239600, 8.60041750175691541318459593531, 9.24808069258600912515827455983, 10.41267650465592522851540130565, 10.9543010850038697713759569335, 11.86961200736979911386764477319, 12.55311265783940088049870684627, 13.48105026248262452381694382029, 14.11072074000488651716000211503, 15.562480791034634579843965506277, 16.36807725980600157606364971638, 16.896779728319556298329184719567, 17.77898633158269627414162894234, 18.67587385482934264266273560680, 19.53985797377097767447970395335, 20.06630565925052102385605547282, 20.66820658301076220928533629940, 21.44595974321537768626355405040

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