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Properties

Label 1-187-187.5-r1-0-0
Degree $1$
Conductor $187$
Sign $-0.977 + 0.208i$
Analytic cond. $20.0959$
Root an. cond. $20.0959$
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  + (0.156 − 0.987i)2-s + (−0.996 − 0.0784i)3-s + (−0.951 − 0.309i)4-s + (0.522 + 0.852i)5-s + (−0.233 + 0.972i)6-s + (0.0784 + 0.996i)7-s + (−0.453 + 0.891i)8-s + (0.987 + 0.156i)9-s + (0.923 − 0.382i)10-s + (0.923 + 0.382i)12-s + (−0.587 − 0.809i)13-s + (0.996 + 0.0784i)14-s + (−0.453 − 0.891i)15-s + (0.809 + 0.587i)16-s + (0.309 − 0.951i)18-s + (−0.891 − 0.453i)19-s + ⋯
L(s)  = 1  + (0.156 − 0.987i)2-s + (−0.996 − 0.0784i)3-s + (−0.951 − 0.309i)4-s + (0.522 + 0.852i)5-s + (−0.233 + 0.972i)6-s + (0.0784 + 0.996i)7-s + (−0.453 + 0.891i)8-s + (0.987 + 0.156i)9-s + (0.923 − 0.382i)10-s + (0.923 + 0.382i)12-s + (−0.587 − 0.809i)13-s + (0.996 + 0.0784i)14-s + (−0.453 − 0.891i)15-s + (0.809 + 0.587i)16-s + (0.309 − 0.951i)18-s + (−0.891 − 0.453i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.977 + 0.208i$
Analytic conductor: \(20.0959\)
Root analytic conductor: \(20.0959\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (1:\ ),\ -0.977 + 0.208i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02801834566 - 0.2654808489i\)
\(L(\frac12)\) \(\approx\) \(0.02801834566 - 0.2654808489i\)
\(L(1)\) \(\approx\) \(0.6273178723 - 0.2402597898i\)
\(L(1)\) \(\approx\) \(0.6273178723 - 0.2402597898i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.156 - 0.987i)T \)
3 \( 1 + (-0.996 - 0.0784i)T \)
5 \( 1 + (0.522 + 0.852i)T \)
7 \( 1 + (0.0784 + 0.996i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (-0.891 - 0.453i)T \)
23 \( 1 + (-0.382 - 0.923i)T \)
29 \( 1 + (0.649 - 0.760i)T \)
31 \( 1 + (0.233 + 0.972i)T \)
37 \( 1 + (0.760 + 0.649i)T \)
41 \( 1 + (-0.649 - 0.760i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (0.156 - 0.987i)T \)
59 \( 1 + (0.891 - 0.453i)T \)
61 \( 1 + (-0.972 - 0.233i)T \)
67 \( 1 - T \)
71 \( 1 + (0.852 - 0.522i)T \)
73 \( 1 + (-0.649 + 0.760i)T \)
79 \( 1 + (-0.852 - 0.522i)T \)
83 \( 1 + (-0.987 + 0.156i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.972 + 0.233i)T \)
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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

−27.37674387865332243136862159546, −26.51263097499184281204491911960, −25.4022395871731423466335865569, −24.33372007133722224342434142300, −23.71081608531149533233116161728, −23.02783770314680212820876051629, −21.75858662100586220015250866640, −21.20586251655781527148753062987, −19.68859492256576570148528618589, −18.26790341594163049352592266903, −17.34103431717708470610838810350, −16.74753810605340848358343255012, −16.184385319776288418132467658315, −14.78440388817087477030953193446, −13.617234132548771656084089467341, −12.851348480192919531299905632480, −11.75938904221764348712686716639, −10.21780707172243351124098693910, −9.40720383121165694793829277797, −7.98004967357316373663863454821, −6.84840753073992616110310484662, −5.92294055352627035895299442390, −4.78376190138736057993499646967, −4.11525144144713237736254251771, −1.32471472958458731838762952981, 0.10718302763845288282224790756, 1.915613471725090658419869891867, 2.9397927769235643438204539284, 4.658021205029325998765862052027, 5.64612938904278679093761965531, 6.61292285700375195980898211443, 8.380229359494842518505501161450, 9.84990200943426483028147415352, 10.482494823514711972322940158623, 11.50301643346758815164189509243, 12.35473866512740150583210809303, 13.2445426536283611282341482774, 14.58462129296514007885116566965, 15.469368661200933831457674930128, 17.1693433691552848787100204380, 17.92321122027290730246011105957, 18.60282698963168077984685522161, 19.495117352416439318851687715546, 21.0263753902909763788527086894, 21.8169751211238176896347940188, 22.34682996609031026374102127729, 23.15584797328988330909042410759, 24.337296585961888943134477768771, 25.47407871448149238504839060613, 26.85514260528453090651773207052

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