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Properties

Label 1-187-187.113-r1-0-0
Degree $1$
Conductor $187$
Sign $-0.907 + 0.420i$
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  + (0.891 − 0.453i)2-s + (0.522 − 0.852i)3-s + (0.587 − 0.809i)4-s + (−0.760 + 0.649i)5-s + (0.0784 − 0.996i)6-s + (−0.852 + 0.522i)7-s + (0.156 − 0.987i)8-s + (−0.453 − 0.891i)9-s + (−0.382 + 0.923i)10-s + (−0.382 − 0.923i)12-s + (−0.951 − 0.309i)13-s + (−0.522 + 0.852i)14-s + (0.156 + 0.987i)15-s + (−0.309 − 0.951i)16-s + (−0.809 − 0.587i)18-s + (−0.987 − 0.156i)19-s + ⋯
L(s)  = 1  + (0.891 − 0.453i)2-s + (0.522 − 0.852i)3-s + (0.587 − 0.809i)4-s + (−0.760 + 0.649i)5-s + (0.0784 − 0.996i)6-s + (−0.852 + 0.522i)7-s + (0.156 − 0.987i)8-s + (−0.453 − 0.891i)9-s + (−0.382 + 0.923i)10-s + (−0.382 − 0.923i)12-s + (−0.951 − 0.309i)13-s + (−0.522 + 0.852i)14-s + (0.156 + 0.987i)15-s + (−0.309 − 0.951i)16-s + (−0.809 − 0.587i)18-s + (−0.987 − 0.156i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.907 + 0.420i$
Analytic conductor: \(20.0959\)
Root analytic conductor: \(20.0959\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (1:\ ),\ -0.907 + 0.420i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2596965980 - 1.178347066i\)
\(L(\frac12)\) \(\approx\) \(-0.2596965980 - 1.178347066i\)
\(L(1)\) \(\approx\) \(1.040487051 - 0.7627310376i\)
\(L(1)\) \(\approx\) \(1.040487051 - 0.7627310376i\)

Euler product

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

−27.17543715262081855181470994593, −26.42959126570936913939035599328, −25.55414550339713195150582732021, −24.58994126186172415680592156759, −23.558895518621317487800813951343, −22.74723801511419019544457815226, −21.79520437813132405723587902182, −20.90272764712318786528490779221, −19.881120928401072886587301022535, −19.42185257300604367243828160027, −17.14338754459750214029816950348, −16.47005467963214379326966921947, −15.72432781622787775898615000746, −14.86353224233046847100526326711, −13.83220671497419187773488755458, −12.81933482940822191361143035644, −11.84837363197156211216185278050, −10.5486406821945957347025674804, −9.28359566757669647728295760570, −8.1248526172144724343257495653, −7.16821960237721584345531286951, −5.65833789240206721900260713945, −4.35076209088783773485548487288, −3.876082664536149893983633834773, −2.51926340950473272853223977637, 0.264373527510762297806560561760, 2.27008980718814730660137306297, 3.00233627467462802805184754442, 4.17064806358420649973925149174, 5.942585778311299452479654904040, 6.79337412181880388789890081852, 7.82523270300446529530232109429, 9.35607096071440308150439082136, 10.60249599653908088828564139439, 11.89629379983196893406060345129, 12.48830731155373555365618911987, 13.402159255557111116726164708966, 14.7021233777467914008587145144, 15.071631410306538860762863284976, 16.35288867795899451163029547291, 18.13304529256217721748990373532, 19.070875617275203146774553987544, 19.57491036725000147705492597187, 20.40983845326270887716015286716, 21.89727673821265483863039917415, 22.53798349050341719789423792515, 23.53887789413189547108785705293, 24.24906064756903157569196988554, 25.32691755679365470813280174648, 26.101244499656191846423158388353

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