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Properties

Label 1-149-149.31-r0-0-0
Degree $1$
Conductor $149$
Sign $-0.966 - 0.255i$
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  + (0.778 − 0.628i)2-s + (−0.828 − 0.559i)3-s + (0.210 − 0.977i)4-s + (0.0424 − 0.999i)5-s + (−0.996 + 0.0848i)6-s + (−0.594 − 0.803i)7-s + (−0.450 − 0.892i)8-s + (0.372 + 0.927i)9-s + (−0.594 − 0.803i)10-s + (0.210 + 0.977i)11-s + (−0.721 + 0.691i)12-s + (−0.127 + 0.991i)13-s + (−0.967 − 0.251i)14-s + (−0.594 + 0.803i)15-s + (−0.911 − 0.411i)16-s + (−0.828 − 0.559i)17-s + ⋯
L(s)  = 1  + (0.778 − 0.628i)2-s + (−0.828 − 0.559i)3-s + (0.210 − 0.977i)4-s + (0.0424 − 0.999i)5-s + (−0.996 + 0.0848i)6-s + (−0.594 − 0.803i)7-s + (−0.450 − 0.892i)8-s + (0.372 + 0.927i)9-s + (−0.594 − 0.803i)10-s + (0.210 + 0.977i)11-s + (−0.721 + 0.691i)12-s + (−0.127 + 0.991i)13-s + (−0.967 − 0.251i)14-s + (−0.594 + 0.803i)15-s + (−0.911 − 0.411i)16-s + (−0.828 − 0.559i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $-0.966 - 0.255i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ -0.966 - 0.255i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1411087572 - 1.085961279i\)
\(L(\frac12)\) \(\approx\) \(0.1411087572 - 1.085961279i\)
\(L(1)\) \(\approx\) \(0.7206010692 - 0.8408893251i\)
\(L(1)\) \(\approx\) \(0.7206010692 - 0.8408893251i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.778 - 0.628i)T \)
3 \( 1 + (-0.828 - 0.559i)T \)
5 \( 1 + (0.0424 - 0.999i)T \)
7 \( 1 + (-0.594 - 0.803i)T \)
11 \( 1 + (0.210 + 0.977i)T \)
13 \( 1 + (-0.127 + 0.991i)T \)
17 \( 1 + (-0.828 - 0.559i)T \)
19 \( 1 + (0.873 - 0.487i)T \)
23 \( 1 + (-0.127 - 0.991i)T \)
29 \( 1 + (0.985 + 0.169i)T \)
31 \( 1 + (-0.127 - 0.991i)T \)
37 \( 1 + (0.210 + 0.977i)T \)
41 \( 1 + (-0.911 - 0.411i)T \)
43 \( 1 + (0.942 - 0.333i)T \)
47 \( 1 + (0.873 + 0.487i)T \)
53 \( 1 + (0.942 + 0.333i)T \)
59 \( 1 + (0.524 - 0.851i)T \)
61 \( 1 + (0.778 - 0.628i)T \)
67 \( 1 + (-0.967 + 0.251i)T \)
71 \( 1 + (0.0424 - 0.999i)T \)
73 \( 1 + (-0.292 - 0.956i)T \)
79 \( 1 + (-0.292 + 0.956i)T \)
83 \( 1 + (-0.292 - 0.956i)T \)
89 \( 1 + (0.660 + 0.750i)T \)
97 \( 1 + (0.942 + 0.333i)T \)
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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

−28.80920755892593390574963179200, −27.25337703792069075387520027308, −26.65105050322207624985849223562, −25.54976314534637848345955145671, −24.601234111231627551588372241779, −23.38287509304784566513131179670, −22.56499867832650079606261553358, −21.93488000038099611442762701747, −21.35910141280282479260291786438, −19.67864659285036868845761824925, −18.20839143493859432703089150011, −17.50073701410695691445165730151, −16.12224336555552845282658687777, −15.57972774201792458061061352374, −14.64987264869332577141682490141, −13.395661849452561777284739317747, −12.17228078959245945330703509189, −11.30786148900449525005012318460, −10.172213454569298782505844851872, −8.69553964925299720647924202486, −7.12993182575862185468060109032, −6.02253321274133484047692529786, −5.51902722810709425207982796158, −3.78354619694189997996195186396, −2.90182580055141637273334932590, 0.8318423284297148312575059396, 2.1883451729559549238635134007, 4.27544239669899372624163384245, 4.88740882759229843030234599804, 6.36326954338233046923949852290, 7.21533321322006221599445422077, 9.266146664472453654235418714615, 10.27836076815579092893564138754, 11.589666478773121401018250943763, 12.30036664837896235736589312895, 13.24341341126469214992311469317, 13.937804178364327312059297189013, 15.70301560610170847851441379833, 16.57946281565572355183673883986, 17.63298149705347669173763636888, 18.908966323642025474242377463957, 19.98209232131659491692890370771, 20.57977506117210789110522480072, 22.03956597087488333662170074838, 22.69444348331725776930529514548, 23.75073673260420104023878168124, 24.239769081907123884913370544772, 25.34041330615883981892037553465, 27.00823932647431208998171372904, 28.2488053775777247821788834047

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