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Properties

Label 1-149-149.7-r0-0-0
Degree $1$
Conductor $149$
Sign $0.592 - 0.805i$
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  + (0.967 − 0.251i)2-s + (−0.985 + 0.169i)3-s + (0.873 − 0.487i)4-s + (0.210 − 0.977i)5-s + (−0.911 + 0.411i)6-s + (0.0424 + 0.999i)7-s + (0.721 − 0.691i)8-s + (0.942 − 0.333i)9-s + (−0.0424 − 0.999i)10-s + (−0.873 − 0.487i)11-s + (−0.778 + 0.628i)12-s + (0.594 − 0.803i)13-s + (0.292 + 0.956i)14-s + (−0.0424 + 0.999i)15-s + (0.524 − 0.851i)16-s + (0.985 − 0.169i)17-s + ⋯
L(s)  = 1  + (0.967 − 0.251i)2-s + (−0.985 + 0.169i)3-s + (0.873 − 0.487i)4-s + (0.210 − 0.977i)5-s + (−0.911 + 0.411i)6-s + (0.0424 + 0.999i)7-s + (0.721 − 0.691i)8-s + (0.942 − 0.333i)9-s + (−0.0424 − 0.999i)10-s + (−0.873 − 0.487i)11-s + (−0.778 + 0.628i)12-s + (0.594 − 0.803i)13-s + (0.292 + 0.956i)14-s + (−0.0424 + 0.999i)15-s + (0.524 − 0.851i)16-s + (0.985 − 0.169i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.592 - 0.805i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.592 - 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.355120060 - 0.6859047349i\)
\(L(\frac12)\) \(\approx\) \(1.355120060 - 0.6859047349i\)
\(L(1)\) \(\approx\) \(1.365826339 - 0.4011221339i\)
\(L(1)\) \(\approx\) \(1.365826339 - 0.4011221339i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.967 - 0.251i)T \)
3 \( 1 + (-0.985 + 0.169i)T \)
5 \( 1 + (0.210 - 0.977i)T \)
7 \( 1 + (0.0424 + 0.999i)T \)
11 \( 1 + (-0.873 - 0.487i)T \)
13 \( 1 + (0.594 - 0.803i)T \)
17 \( 1 + (0.985 - 0.169i)T \)
19 \( 1 + (-0.828 - 0.559i)T \)
23 \( 1 + (0.594 + 0.803i)T \)
29 \( 1 + (0.660 + 0.750i)T \)
31 \( 1 + (-0.594 - 0.803i)T \)
37 \( 1 + (0.873 + 0.487i)T \)
41 \( 1 + (-0.524 + 0.851i)T \)
43 \( 1 + (0.127 + 0.991i)T \)
47 \( 1 + (-0.828 + 0.559i)T \)
53 \( 1 + (-0.127 + 0.991i)T \)
59 \( 1 + (-0.372 - 0.927i)T \)
61 \( 1 + (-0.967 + 0.251i)T \)
67 \( 1 + (-0.292 + 0.956i)T \)
71 \( 1 + (-0.210 + 0.977i)T \)
73 \( 1 + (-0.996 - 0.0848i)T \)
79 \( 1 + (0.996 - 0.0848i)T \)
83 \( 1 + (0.996 + 0.0848i)T \)
89 \( 1 + (0.450 + 0.892i)T \)
97 \( 1 + (0.127 - 0.991i)T \)
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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

−28.73038059182363438395033860938, −27.15944857091605270514312901591, −26.14748643998768439661285086134, −25.286464841029032416054922744016, −23.804428713696575072480735967672, −23.253231863463501389168781860345, −22.75237438828610890060290461428, −21.4480933577722356757547907272, −20.88988648063199963622169995224, −19.225149131423892217652451455340, −18.14510351928213711606790924169, −17.012129066831423166431854550, −16.26196627319735499045655622256, −15.023064811851295126410535981, −13.99398281848315634602660587067, −13.05177562487490164097413947743, −11.96640812270758908856609471729, −10.74316037995190238578629697032, −10.35758212895974695111077609359, −7.75755052330057505762577016622, −6.88048854960783072325391770480, −6.07180157546898776956541621888, −4.76519038368892364770048098954, −3.625244078204336960278411184559, −1.926554366663360921864916686266, 1.26952000196077954611933677008, 3.02706902075019226790687815970, 4.698664748705859212363099079920, 5.47450140691281192564484476473, 6.14693087707584400177527754852, 7.958816635179951375928686204676, 9.53966589965032774515877252693, 10.77955611990838496707293908570, 11.71138835453491969714332261047, 12.75704641973517727956528680355, 13.21703694095641845252666354686, 15.01416021751386189656971597832, 15.86003226540234433427241984618, 16.607392975101784214910259329986, 17.95948111789718472350133441305, 19.04106726833597192383422472420, 20.50103943409362613049573309762, 21.37218290657413969599711823759, 21.834057991471589884162474336330, 23.23977332255684654903883216544, 23.71052626714756272096500839610, 24.79531844764591792297347532029, 25.59277274428432137774197517204, 27.67294603545412919922302542292, 28.0518596455985589348208602584

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