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Properties

Label 1-837-837.626-r0-0-0
Degree $1$
Conductor $837$
Sign $0.975 - 0.218i$
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  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (0.939 + 0.342i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.173 − 0.984i)11-s + (−0.766 + 0.642i)13-s + (0.939 + 0.342i)14-s + (0.766 + 0.642i)16-s + 17-s + (−0.5 − 0.866i)19-s + (−0.766 − 0.642i)20-s + (0.939 + 0.342i)22-s + (−0.939 − 0.342i)23-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (0.939 + 0.342i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.173 − 0.984i)11-s + (−0.766 + 0.642i)13-s + (0.939 + 0.342i)14-s + (0.766 + 0.642i)16-s + 17-s + (−0.5 − 0.866i)19-s + (−0.766 − 0.642i)20-s + (0.939 + 0.342i)22-s + (−0.939 − 0.342i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.975 - 0.218i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.975 - 0.218i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.262673365 - 0.1397594000i\)
\(L(\frac12)\) \(\approx\) \(1.262673365 - 0.1397594000i\)
\(L(1)\) \(\approx\) \(0.9980794016 + 0.2079979271i\)
\(L(1)\) \(\approx\) \(0.9980794016 + 0.2079979271i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.173 - 0.984i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−22.05408615056506023203125623402, −21.27495334356916425894145407685, −20.64543963511145451337496773225, −19.93889641199168646390506153481, −18.923423541746673144029129209737, −18.25750497815148674967401134291, −17.48797002578622429388001308833, −16.996854123745203493470200432969, −15.74174135176496976848079976250, −14.52508276087399788948529458239, −14.16790122500905106193948911205, −12.76405625169480689396340964746, −12.46986904659584635134261465431, −11.79797398202140579800142522434, −10.34310145485694501949904344441, −10.02078390187295747155524109720, −9.13309388179779178466313069667, −8.37087210186208637479321255406, −7.34607267663488376011238234603, −5.773898207366031320320086137907, −5.28568159630427817746276427857, −4.247624696663920737691880191446, −2.96006153310527159060866146942, −2.10913959023321485305204227026, −1.38798592728350213443666910123, 0.636007448314903817655589418700, 1.94994920838553130998553205977, 3.41134050004081900602861095790, 4.4561853907366635294825950084, 5.38898528896322802559287958860, 6.3130853926589547653031345650, 6.94111050481966556149256453272, 7.86490556500027670076464863486, 8.78975377053001489311680553354, 9.78079518207068097679001011503, 10.280792906741953459039199809959, 11.30714475677085366840150774831, 12.6341406765122528259809943856, 13.70224029134034807691340401889, 14.01264301005659218895099145855, 14.66561540178856426390898387125, 15.797834192780007937713324247550, 16.78292917574619964981373727367, 17.09609802348892101535075448478, 17.85492405821431681984396714077, 18.891258283751787845648706243352, 19.3624790467114752409515118581, 20.622724212973293223393470938703, 21.60021842534667332037054393461, 22.0955454926169907411205853736

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