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Properties

Label 1-837-837.328-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.203 - 0.979i$
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  + (0.848 − 0.529i)2-s + (0.438 − 0.898i)4-s + (0.766 − 0.642i)5-s + (0.961 + 0.275i)7-s + (−0.104 − 0.994i)8-s + (0.309 − 0.951i)10-s + (−0.719 + 0.694i)11-s + (−0.882 + 0.469i)13-s + (0.961 − 0.275i)14-s + (−0.615 − 0.788i)16-s + (−0.104 − 0.994i)17-s + (0.309 − 0.951i)19-s + (−0.241 − 0.970i)20-s + (−0.241 + 0.970i)22-s + (−0.719 − 0.694i)23-s + ⋯
L(s)  = 1  + (0.848 − 0.529i)2-s + (0.438 − 0.898i)4-s + (0.766 − 0.642i)5-s + (0.961 + 0.275i)7-s + (−0.104 − 0.994i)8-s + (0.309 − 0.951i)10-s + (−0.719 + 0.694i)11-s + (−0.882 + 0.469i)13-s + (0.961 − 0.275i)14-s + (−0.615 − 0.788i)16-s + (−0.104 − 0.994i)17-s + (0.309 − 0.951i)19-s + (−0.241 − 0.970i)20-s + (−0.241 + 0.970i)22-s + (−0.719 − 0.694i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.203 - 0.979i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.203 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.774338649 - 2.180161564i\)
\(L(\frac12)\) \(\approx\) \(1.774338649 - 2.180161564i\)
\(L(1)\) \(\approx\) \(1.673096927 - 0.9795591341i\)
\(L(1)\) \(\approx\) \(1.673096927 - 0.9795591341i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.848 - 0.529i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (0.961 + 0.275i)T \)
11 \( 1 + (-0.719 + 0.694i)T \)
13 \( 1 + (-0.882 + 0.469i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.719 - 0.694i)T \)
29 \( 1 + (0.848 - 0.529i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.990 + 0.139i)T \)
43 \( 1 + (0.848 - 0.529i)T \)
47 \( 1 + (0.990 - 0.139i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (-0.882 + 0.469i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (0.438 + 0.898i)T \)
83 \( 1 + (0.0348 + 0.999i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (0.961 + 0.275i)T \)
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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

−22.30429688464828948269587234320, −21.551492651255638098995355100048, −21.186655213808600588355926563160, −20.256828518129859469515488432290, −19.19022076824072725897287123757, −17.95722850169733989808709275800, −17.611444961806923920186388615556, −16.74982208597895336622141735658, −15.77843662566893684064676118459, −14.903012159661918566678482862323, −14.22620157231774437800253997324, −13.76895861735468156419578384797, −12.7513039325397001537580017989, −11.95853636965465589630947278698, −10.780809823712065795597047030361, −10.43271604415646885859289329755, −8.97699773586761901547488074425, −7.78719106582493422950712052161, −7.51382282417874986391825203558, −6.09474330148154147923822505232, −5.65889216159609655882011197148, −4.70430860496828981790655962082, −3.596304891608252590176875398756, −2.63842919660746095865489903127, −1.69980457700941512328391320163, 0.951257623739289821219736945470, 2.30351314624191287891175266501, 2.4748850402803538599103768142, 4.379909581740380270528082051858, 4.86236494121202966620738707625, 5.49027293901431019798124489730, 6.66073946853425548588672214072, 7.62890691574231312740777741118, 8.888465950404953959663824184207, 9.72844965769890005011047487349, 10.43855733742298400724044200039, 11.54844320500030038095607347139, 12.16915759222110240518429970336, 12.92967266528523922787933736742, 13.91452127455008382241669142875, 14.29494618822681827269308825269, 15.39054024461725567403225372619, 16.039802782735296695305680900086, 17.26871986863763877902089837629, 17.93390254384533865372081285522, 18.74808613892131514797757950387, 19.9223897774879034546900212901, 20.52866444370143308693741103765, 21.09696438209949522638597957273, 21.82912094952590333664302591529

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