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Properties

Label 1-837-837.376-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.0957 - 0.995i$
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  + (0.961 − 0.275i)2-s + (0.848 − 0.529i)4-s + (0.173 − 0.984i)5-s + (−0.374 − 0.927i)7-s + (0.669 − 0.743i)8-s + (−0.104 − 0.994i)10-s + (0.990 + 0.139i)11-s + (0.961 + 0.275i)13-s + (−0.615 − 0.788i)14-s + (0.438 − 0.898i)16-s + (−0.978 − 0.207i)17-s + (0.913 + 0.406i)19-s + (−0.374 − 0.927i)20-s + (0.990 − 0.139i)22-s + (−0.374 + 0.927i)23-s + ⋯
L(s)  = 1  + (0.961 − 0.275i)2-s + (0.848 − 0.529i)4-s + (0.173 − 0.984i)5-s + (−0.374 − 0.927i)7-s + (0.669 − 0.743i)8-s + (−0.104 − 0.994i)10-s + (0.990 + 0.139i)11-s + (0.961 + 0.275i)13-s + (−0.615 − 0.788i)14-s + (0.438 − 0.898i)16-s + (−0.978 − 0.207i)17-s + (0.913 + 0.406i)19-s + (−0.374 − 0.927i)20-s + (0.990 − 0.139i)22-s + (−0.374 + 0.927i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0957 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0957 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.0957 - 0.995i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.0957 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.990694755 - 2.191407651i\)
\(L(\frac12)\) \(\approx\) \(1.990694755 - 2.191407651i\)
\(L(1)\) \(\approx\) \(1.779273906 - 0.9459847466i\)
\(L(1)\) \(\approx\) \(1.779273906 - 0.9459847466i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.961 - 0.275i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (-0.374 - 0.927i)T \)
11 \( 1 + (0.990 + 0.139i)T \)
13 \( 1 + (0.961 + 0.275i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.374 + 0.927i)T \)
29 \( 1 + (0.961 - 0.275i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.559 - 0.829i)T \)
43 \( 1 + (-0.719 - 0.694i)T \)
47 \( 1 + (-0.997 + 0.0697i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.719 + 0.694i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (0.0348 - 0.999i)T \)
83 \( 1 + (0.961 - 0.275i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (0.990 + 0.139i)T \)
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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

−22.47561250499792545953118396468, −21.78599601778731608746011513037, −21.10617229984473670965776460046, −19.937008482885879711892499041940, −19.372418041835947301348584676364, −18.1523104904016824154794431901, −17.73230571475836028222331401506, −16.34847111590567157112672263819, −15.81432169722653680693350266599, −14.97216307310514722977168720034, −14.34699757619554823666174130529, −13.53580706999371403201891244400, −12.73411486062401837081496558886, −11.69913378586757773692363512797, −11.23447447708401500532949127456, −10.211019198291337988601369632553, −9.0190793150635819508293843841, −8.1606126164182932617548673473, −6.84563544569911747677772782215, −6.4067251499749602779247830590, −5.6936351232266957224108832096, −4.48678638410475998392723732195, −3.42831583559591630998478787807, −2.79396797935966790800730680329, −1.734455414921975595671380645547, 1.05110020876703768714174528785, 1.74053771531306329600438393871, 3.310307146960975694533470973143, 4.06201846856263863148403739478, 4.74417955824885299891992721464, 5.879477498910218006692650259897, 6.60312083856540826454245526319, 7.54705279377776738840211162165, 8.81261830518629310827871311825, 9.67957686514247342218571645079, 10.52715098595924632648472413430, 11.64409636447632084154771787974, 12.07205757316157017156373702906, 13.35992160979214986201245162311, 13.523667423572658290807245237271, 14.34184048581409298503147046082, 15.659222768558216179077876895194, 16.099814379413818656114783419780, 16.94739121781282761244472762744, 17.76736835963756413459385238422, 19.12578048369262570508646337802, 19.9244537436282084405942649985, 20.32916047409410914537950092853, 21.069958560462672544244466904601, 21.98469785159185693500044425092

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