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Properties

Label 1-149-149.42-r0-0-0
Degree $1$
Conductor $149$
Sign $-0.958 - 0.286i$
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  + (−0.942 − 0.333i)2-s + (0.292 + 0.956i)3-s + (0.778 + 0.628i)4-s + (−0.721 − 0.691i)5-s + (0.0424 − 0.999i)6-s + (−0.450 − 0.892i)7-s + (−0.524 − 0.851i)8-s + (−0.828 + 0.559i)9-s + (0.450 + 0.892i)10-s + (−0.778 + 0.628i)11-s + (−0.372 + 0.927i)12-s + (−0.660 + 0.750i)13-s + (0.127 + 0.991i)14-s + (0.450 − 0.892i)15-s + (0.210 + 0.977i)16-s + (−0.292 − 0.956i)17-s + ⋯
L(s)  = 1  + (−0.942 − 0.333i)2-s + (0.292 + 0.956i)3-s + (0.778 + 0.628i)4-s + (−0.721 − 0.691i)5-s + (0.0424 − 0.999i)6-s + (−0.450 − 0.892i)7-s + (−0.524 − 0.851i)8-s + (−0.828 + 0.559i)9-s + (0.450 + 0.892i)10-s + (−0.778 + 0.628i)11-s + (−0.372 + 0.927i)12-s + (−0.660 + 0.750i)13-s + (0.127 + 0.991i)14-s + (0.450 − 0.892i)15-s + (0.210 + 0.977i)16-s + (−0.292 − 0.956i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $-0.958 - 0.286i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ -0.958 - 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007809892613 - 0.05347065318i\)
\(L(\frac12)\) \(\approx\) \(0.007809892613 - 0.05347065318i\)
\(L(1)\) \(\approx\) \(0.4336248296 + 0.003865003143i\)
\(L(1)\) \(\approx\) \(0.4336248296 + 0.003865003143i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.942 - 0.333i)T \)
3 \( 1 + (0.292 + 0.956i)T \)
5 \( 1 + (-0.721 - 0.691i)T \)
7 \( 1 + (-0.450 - 0.892i)T \)
11 \( 1 + (-0.778 + 0.628i)T \)
13 \( 1 + (-0.660 + 0.750i)T \)
17 \( 1 + (-0.292 - 0.956i)T \)
19 \( 1 + (-0.967 - 0.251i)T \)
23 \( 1 + (-0.660 - 0.750i)T \)
29 \( 1 + (-0.996 + 0.0848i)T \)
31 \( 1 + (0.660 + 0.750i)T \)
37 \( 1 + (0.778 - 0.628i)T \)
41 \( 1 + (-0.210 - 0.977i)T \)
43 \( 1 + (-0.985 - 0.169i)T \)
47 \( 1 + (-0.967 + 0.251i)T \)
53 \( 1 + (0.985 - 0.169i)T \)
59 \( 1 + (-0.873 - 0.487i)T \)
61 \( 1 + (0.942 + 0.333i)T \)
67 \( 1 + (-0.127 + 0.991i)T \)
71 \( 1 + (0.721 + 0.691i)T \)
73 \( 1 + (-0.594 - 0.803i)T \)
79 \( 1 + (0.594 - 0.803i)T \)
83 \( 1 + (0.594 + 0.803i)T \)
89 \( 1 + (0.911 - 0.411i)T \)
97 \( 1 + (-0.985 + 0.169i)T \)
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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

−28.481167028416207763501200693590, −27.564706422884782076606915957046, −26.32587354048067057193272636544, −25.82009174519344684265123743130, −24.74378973867734651591193609711, −23.927220181758461578148477884749, −23.02996118830873037888295447695, −21.63912966439212878354486478849, −20.04801373464057066013935915321, −19.3075025992578278679364871394, −18.66226224627112492639957185104, −17.91707563865554481534300624436, −16.660186068558499753630292225435, −15.21301694909178023647646584979, −14.95524029187867009449227110344, −13.222881798687076571720174863902, −12.05912449243709343923244172185, −11.07518611596422804591429336759, −9.819525002359993444974711110890, −8.28706862642351313168429566201, −7.92993979654017620454745089116, −6.57584918980002606947458489463, −5.76586049866398552984593276370, −3.13569578393257746910245660170, −2.150703947707392318635739003585, 0.05498611903920108111699702369, 2.37048697575833929849084788808, 3.832912455832024923958810343780, 4.7743878505834537389396589606, 6.954301687184590800663459052027, 7.98860707679347980614518370152, 9.07882774794974886175181234344, 9.96145350733923389814120156279, 10.892051452462461976506460544841, 12.0279720723583979885707837870, 13.23014829176543243799568390120, 14.855597706712695196519988072569, 16.00210492469553122028420604074, 16.48476708394801702639967240103, 17.450337117784110664184983984211, 18.99197424754639963264582806823, 19.94097950090777071128425554105, 20.435579087709183578835483431871, 21.34405074492376923361426780775, 22.65029430719508805694248039899, 23.804315121933236607158297948277, 25.055971506974515629941851010151, 26.246851310693753818619418243243, 26.64402383650064994893740139076, 27.63785471315603717514165069537

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