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Properties

Label 1-837-837.25-r0-0-0
Degree $1$
Conductor $837$
Sign $0.941 + 0.336i$
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  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.939 + 0.342i)5-s + (0.173 + 0.984i)7-s + (−0.5 − 0.866i)8-s + 10-s + (0.766 − 0.642i)11-s + (0.766 + 0.642i)13-s + (0.173 − 0.984i)14-s + (0.173 + 0.984i)16-s + (−0.5 − 0.866i)17-s + 19-s + (−0.939 − 0.342i)20-s + (−0.939 + 0.342i)22-s + (0.766 + 0.642i)23-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.939 + 0.342i)5-s + (0.173 + 0.984i)7-s + (−0.5 − 0.866i)8-s + 10-s + (0.766 − 0.642i)11-s + (0.766 + 0.642i)13-s + (0.173 − 0.984i)14-s + (0.173 + 0.984i)16-s + (−0.5 − 0.866i)17-s + 19-s + (−0.939 − 0.342i)20-s + (−0.939 + 0.342i)22-s + (0.766 + 0.642i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.941 + 0.336i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.941 + 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8489505663 + 0.1470163441i\)
\(L(\frac12)\) \(\approx\) \(0.8489505663 + 0.1470163441i\)
\(L(1)\) \(\approx\) \(0.7029742699 + 0.02529375369i\)
\(L(1)\) \(\approx\) \(0.7029742699 + 0.02529375369i\)

Euler product

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

−22.36985655594393476319060014484, −20.6672914132260082989558372359, −20.41762081611362858560283477025, −19.70982827393918287960420109325, −18.96099078869058071211029116261, −18.00656368660727966120165904852, −17.1994921088010500340192336643, −16.65124958006313540349739893330, −15.74935249694600326573120845691, −15.09395494338983748961113960758, −14.287235933312138068331929601869, −13.109388001831629424333566932382, −12.151085688409585933842978762707, −11.15179852845774017390298560422, −10.70417753181610048377553283802, −9.62229476342437667080699921766, −8.73055694373589949218106163477, −7.97394281923007115956536690519, −7.224982671048597193778320932294, −6.51381784674423320073112438414, −5.23481192792446386939658980676, −4.179257905063979480750666340511, −3.25735682315132805421376123810, −1.595561676923372849998887475108, −0.78675767362291799110477346067, 0.89815087096709033947706702344, 2.14191739260985657538646296384, 3.22030718420118154115703738654, 3.902920222063182911962021770322, 5.39887795823620853036034425153, 6.56653896648402061548764018937, 7.26704479943317620485421938631, 8.27904923713850763857875804434, 8.9710758327981409214480274719, 9.57223721442748707801552684810, 11.03326365444887249548157859688, 11.48248323174781420021581702431, 11.89046385820808520381551116504, 13.06237304201520521412816276266, 14.22152717428834069561485852820, 15.23331370025244735895520025236, 15.92521304763194528431659760380, 16.454473244964072691798069820533, 17.59412346895820953818533983439, 18.46975294053079577417685371535, 18.868061709331462198349839140793, 19.61224956189550135942016492074, 20.429289906795806346398963699, 21.283351407050322270510765165996, 22.07001212774029348704117350384

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