Properties

Label 1-837-837.178-r1-0-0
Degree $1$
Conductor $837$
Sign $0.314 + 0.949i$
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  + (0.438 − 0.898i)2-s + (−0.615 − 0.788i)4-s + (0.766 + 0.642i)5-s + (0.0348 − 0.999i)7-s + (−0.978 + 0.207i)8-s + (0.913 − 0.406i)10-s + (−0.848 − 0.529i)11-s + (−0.438 − 0.898i)13-s + (−0.882 − 0.469i)14-s + (−0.241 + 0.970i)16-s + (−0.669 − 0.743i)17-s + (−0.104 + 0.994i)19-s + (0.0348 − 0.999i)20-s + (−0.848 + 0.529i)22-s + (−0.0348 − 0.999i)23-s + ⋯
L(s)  = 1  + (0.438 − 0.898i)2-s + (−0.615 − 0.788i)4-s + (0.766 + 0.642i)5-s + (0.0348 − 0.999i)7-s + (−0.978 + 0.207i)8-s + (0.913 − 0.406i)10-s + (−0.848 − 0.529i)11-s + (−0.438 − 0.898i)13-s + (−0.882 − 0.469i)14-s + (−0.241 + 0.970i)16-s + (−0.669 − 0.743i)17-s + (−0.104 + 0.994i)19-s + (0.0348 − 0.999i)20-s + (−0.848 + 0.529i)22-s + (−0.0348 − 0.999i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.314 + 0.949i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.314 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03460104763 + 0.02498056055i\)
\(L(\frac12)\) \(\approx\) \(0.03460104763 + 0.02498056055i\)
\(L(1)\) \(\approx\) \(0.8653759865 - 0.5943594468i\)
\(L(1)\) \(\approx\) \(0.8653759865 - 0.5943594468i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.438 - 0.898i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (0.0348 - 0.999i)T \)
11 \( 1 + (-0.848 - 0.529i)T \)
13 \( 1 + (-0.438 - 0.898i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.0348 - 0.999i)T \)
29 \( 1 + (-0.438 + 0.898i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.719 + 0.694i)T \)
43 \( 1 + (0.997 - 0.0697i)T \)
47 \( 1 + (0.961 - 0.275i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.997 - 0.0697i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (-0.990 - 0.139i)T \)
83 \( 1 + (-0.438 + 0.898i)T \)
89 \( 1 + (-0.669 + 0.743i)T \)
97 \( 1 + (0.848 + 0.529i)T \)
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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

−21.78348013034467109288232501833, −21.3244346428612987982852321621, −20.4842901682975088513547888826, −19.22757220044979674748172284478, −18.35202442265738513810980355038, −17.449903922747720333445803607636, −17.15333216140604962836119489163, −15.90662766611661019895843211781, −15.4657343945054836067155115544, −14.62645756262471388620810525011, −13.585250198107234741130812755452, −13.048203455403262950844320290161, −12.29545243391640116615139691513, −11.40083864026811037750834744268, −9.89145676101718442938941944428, −9.18171424605334794235297121315, −8.54408011967888818155103335027, −7.52998299880056017199900782111, −6.5062042569912071489454429938, −5.71080676619690780134681764963, −4.974410698917674316277414774285, −4.24577759503263203493115155627, −2.69794532708082803480546505346, −1.908617814627731432673282239949, −0.008004665668316590179141763301, 1.08828475278889697686418373664, 2.365270963978442936899417251617, 3.01495585213849720552706690215, 4.05394384913544404800282936519, 5.13973875710034150818322034151, 5.87905410331755622517239314974, 6.9367842355194582117001970027, 7.980965752198364950892730673737, 9.19231744018471371452907069202, 10.15415409421635096495822356818, 10.60694616113633509568768628636, 11.195155915355321848970499420476, 12.55550691256505550363036853143, 13.12459078182108200542473533611, 13.98340830246546296607343232151, 14.42976497134835208597172572777, 15.44380204023932068467939103909, 16.593534901459648384204519723370, 17.56078709923687029195524775763, 18.31325014029173155677806301012, 18.82183386869931907976604219481, 20.05433621841863594190338228692, 20.45844190534960476569871877556, 21.28432085281016505661104185468, 22.092606241164794565939807246518

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