Properties

Label 1-837-837.439-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.832 - 0.554i$
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  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.173 − 0.984i)5-s + (0.766 − 0.642i)7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.766 − 0.642i)11-s + (−0.939 + 0.342i)13-s + (0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + 17-s + (−0.5 + 0.866i)19-s + (−0.939 − 0.342i)20-s + (0.173 − 0.984i)22-s + (0.173 − 0.984i)23-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.173 − 0.984i)5-s + (0.766 − 0.642i)7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.766 − 0.642i)11-s + (−0.939 + 0.342i)13-s + (0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + 17-s + (−0.5 + 0.866i)19-s + (−0.939 − 0.342i)20-s + (0.173 − 0.984i)22-s + (0.173 − 0.984i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.832 - 0.554i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7164032213 - 2.366577744i\)
\(L(\frac12)\) \(\approx\) \(0.7164032213 - 2.366577744i\)
\(L(1)\) \(\approx\) \(1.262955082 - 1.197168487i\)
\(L(1)\) \(\approx\) \(1.262955082 - 1.197168487i\)

Euler product

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

−22.29510303586727156197338239050, −21.890588090288013961932259073278, −21.27438883097074278568880704310, −20.20202815051429247774741331118, −19.27281218013752961236449684089, −18.229706269662081260898494625389, −17.48740384809748547114664597196, −17.03431470810617387508801439242, −15.67648110142082811502475768616, −14.99691229967273489626814273116, −14.57840632125843361446598028966, −13.864615898767641870384945545121, −12.73071114792179478468697202571, −11.91470234970869204990149245837, −11.348642785509151814693711394410, −10.165571221552368099138014034544, −9.15755199857711849658532129791, −8.07873688408300397883922785685, −7.25148031888568184071711822490, −6.62434558203911817553610067968, −5.51857472988535838410466536184, −4.884712921378393173968623300978, −3.72054180613342975527503626030, −2.75884439184929918256133029098, −1.899527526652017736105209562139, 0.903628195971146250049263222798, 1.61559302200173796670300934261, 2.84693413242846643177102113155, 4.19498099998569876177011106883, 4.520065994773151915111676748421, 5.590776753404765335187089957561, 6.410298434947338769589038998571, 7.69803286223747608061778927979, 8.63211152550237121606460084259, 9.67722880323739031701028501086, 10.34909157774164370750109604404, 11.43203152384722709920004374874, 12.080341330299185835672597349875, 12.75370730359761210938914897287, 13.78895759994035645482325925354, 14.335095404116009608790716561475, 14.99554694439351286597398724624, 16.50039474743228139392972874225, 16.71045362690284975913176312767, 17.81570065548984854666936269636, 18.984265916401281381763496029, 19.59258904490344900374157703329, 20.422889613647452186008327760997, 21.06332429880724235281795895566, 21.57547654416120747581271481876

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