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Properties

Label 1-988-988.7-r0-0-0
Degree $1$
Conductor $988$
Sign $0.757 - 0.652i$
Analytic cond. $4.58825$
Root an. cond. $4.58825$
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  − 3-s + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)7-s + 9-s + (0.866 − 0.5i)11-s + (0.866 + 0.5i)15-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)21-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s − 27-s + (−0.5 + 0.866i)29-s − i·31-s + (−0.866 + 0.5i)33-s + (0.5 + 0.866i)35-s + ⋯
L(s)  = 1  − 3-s + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)7-s + 9-s + (0.866 − 0.5i)11-s + (0.866 + 0.5i)15-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)21-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s − 27-s + (−0.5 + 0.866i)29-s − i·31-s + (−0.866 + 0.5i)33-s + (0.5 + 0.866i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 988 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 988 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(988\)    =    \(2^{2} \cdot 13 \cdot 19\)
Sign: $0.757 - 0.652i$
Analytic conductor: \(4.58825\)
Root analytic conductor: \(4.58825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{988} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 988,\ (0:\ ),\ 0.757 - 0.652i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6555294941 - 0.2432549504i\)
\(L(\frac12)\) \(\approx\) \(0.6555294941 - 0.2432549504i\)
\(L(1)\) \(\approx\) \(0.6407490551 - 0.09448339644i\)
\(L(1)\) \(\approx\) \(0.6407490551 - 0.09448339644i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
19 \( 1 \)
good3 \( 1 - T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−22.20172318742891547876774812059, −21.11879686788886484557786584579, −20.09700745466738600287574283988, −19.23310500674192055770694391044, −18.69125518242064784027506144841, −17.92774668917062098414115555345, −16.99466045375226173870988109128, −16.15765358448222316224462848320, −15.71114518059327438140561378123, −14.82187555406535375276326539984, −13.885498335085061527811204777395, −12.57222257298737172774004973197, −12.17666428748590360009565716018, −11.52903687222301145919741960799, −10.57879852953583287707970699902, −9.81412561907539831387584847749, −8.93371943976980795634576715252, −7.65115277191651484772413152932, −6.86787482967639584949505248873, −6.29956708534149241598225198817, −5.259634875183633241226601052488, −4.233132846811689045277527119807, −3.45761242692842931614418914346, −2.246816242060778388570346276137, −0.7100949322066447535854726362, 0.587182449491989407695953032453, 1.58603063949316848388097431424, 3.63202408913982477087896882665, 3.81080615160550132473042492470, 5.05528994036257362143286611030, 5.95274529775119897260078120769, 6.77483310309046506423579967338, 7.55811131251189074542865460212, 8.59485666129530488925433742801, 9.6184515964301660029355776896, 10.38257505515708670059832882746, 11.37812795725661388609433684587, 11.887738481901693048391728619268, 12.76251927012094386025618275635, 13.34755056235762674250933358096, 14.58306921412218060617303166378, 15.583969893962207141034969767809, 16.20857030405584027155973952514, 16.89728674420590625625276091659, 17.31722882091555107388068053796, 18.66549156314433258110511728203, 19.201005884212353161684304848, 19.92924451759231855691034224372, 20.78026275912770878164560457271, 21.97628683828366081040800271042

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