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Properties

Label 1-149-149.26-r0-0-0
Degree $1$
Conductor $149$
Sign $0.367 - 0.930i$
Analytic cond. $0.691953$
Root an. cond. $0.691953$
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  + (−0.660 + 0.750i)2-s + (−0.0424 − 0.999i)3-s + (−0.127 − 0.991i)4-s + (0.942 − 0.333i)5-s + (0.778 + 0.628i)6-s + (0.372 − 0.927i)7-s + (0.828 + 0.559i)8-s + (−0.996 + 0.0848i)9-s + (−0.372 + 0.927i)10-s + (0.127 − 0.991i)11-s + (−0.985 + 0.169i)12-s + (−0.524 + 0.851i)13-s + (0.450 + 0.892i)14-s + (−0.372 − 0.927i)15-s + (−0.967 + 0.251i)16-s + (0.0424 + 0.999i)17-s + ⋯
L(s)  = 1  + (−0.660 + 0.750i)2-s + (−0.0424 − 0.999i)3-s + (−0.127 − 0.991i)4-s + (0.942 − 0.333i)5-s + (0.778 + 0.628i)6-s + (0.372 − 0.927i)7-s + (0.828 + 0.559i)8-s + (−0.996 + 0.0848i)9-s + (−0.372 + 0.927i)10-s + (0.127 − 0.991i)11-s + (−0.985 + 0.169i)12-s + (−0.524 + 0.851i)13-s + (0.450 + 0.892i)14-s + (−0.372 − 0.927i)15-s + (−0.967 + 0.251i)16-s + (0.0424 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.367 - 0.930i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.367 - 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7007731025 - 0.4765919210i\)
\(L(\frac12)\) \(\approx\) \(0.7007731025 - 0.4765919210i\)
\(L(1)\) \(\approx\) \(0.8159715206 - 0.2171545343i\)
\(L(1)\) \(\approx\) \(0.8159715206 - 0.2171545343i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.660 + 0.750i)T \)
3 \( 1 + (-0.0424 - 0.999i)T \)
5 \( 1 + (0.942 - 0.333i)T \)
7 \( 1 + (0.372 - 0.927i)T \)
11 \( 1 + (0.127 - 0.991i)T \)
13 \( 1 + (-0.524 + 0.851i)T \)
17 \( 1 + (0.0424 + 0.999i)T \)
19 \( 1 + (-0.594 - 0.803i)T \)
23 \( 1 + (-0.524 - 0.851i)T \)
29 \( 1 + (0.210 - 0.977i)T \)
31 \( 1 + (0.524 + 0.851i)T \)
37 \( 1 + (-0.127 + 0.991i)T \)
41 \( 1 + (0.967 - 0.251i)T \)
43 \( 1 + (0.911 - 0.411i)T \)
47 \( 1 + (-0.594 + 0.803i)T \)
53 \( 1 + (-0.911 - 0.411i)T \)
59 \( 1 + (0.292 - 0.956i)T \)
61 \( 1 + (0.660 - 0.750i)T \)
67 \( 1 + (-0.450 + 0.892i)T \)
71 \( 1 + (-0.942 + 0.333i)T \)
73 \( 1 + (-0.721 + 0.691i)T \)
79 \( 1 + (0.721 + 0.691i)T \)
83 \( 1 + (0.721 - 0.691i)T \)
89 \( 1 + (-0.873 + 0.487i)T \)
97 \( 1 + (0.911 + 0.411i)T \)
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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

−28.00434190630210235232589424146, −27.65807884578125155890738735136, −26.52300194246770942471374670875, −25.45697635401337879860723678360, −25.05716654223972268980772259574, −22.737071408379501834416148552256, −22.186809867094557106486804632612, −21.20109587356642548732697515145, −20.64515657862723131382954721003, −19.48763151760603428715340289273, −18.00163963456633659528028803314, −17.67735572489738432973473250295, −16.447142666593447950907442301634, −15.21865702571285548944820341730, −14.2381989151093824281988742086, −12.701459129407581865955307400573, −11.68764485804448167526487680185, −10.51157979842369420553525814288, −9.72897533665795399115765978964, −9.01588552746924319584078696942, −7.621807228415705738242174025881, −5.7929379667870634112853945057, −4.62595241557298143155707150211, −3.00681656851156113144886255135, −2.04058563358809635261393033343, 0.957052489555993838099864315500, 2.14999029586840011369617347939, 4.66124654812152630035351994195, 6.096928536023072125424818369642, 6.716632338236460327946838772, 8.049017250271081145108969303515, 8.858814853864999501489583957200, 10.22660363648368278043713337717, 11.2878053968713549809549422129, 12.936650518582840301374049408391, 13.95670925408484426877106086491, 14.41202577048231174283833114921, 16.28354467917091781347887840163, 17.28330534432338383159957633894, 17.50751611964931284060046850829, 18.93067544650435711769936490024, 19.55180419356197861154559334299, 20.82740573390833763079022247563, 22.19292882143468074009466198065, 23.67689044018032122640397667598, 24.10262738032199847267458444025, 24.86710542621916654366983206010, 26.02907206310686030411459255453, 26.575833055057008494184900018692, 28.03285792406206653890537199706

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