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Properties

Label 1-149-149.148-r0-0-0
Degree $1$
Conductor $149$
Sign $1$
Analytic cond. $0.691953$
Root an. cond. $0.691953$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 21-s + 22-s − 23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 21-s + 22-s − 23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $1$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{149} (148, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6703958822\)
\(L(\frac12)\) \(\approx\) \(0.6703958822\)
\(L(1)\) \(\approx\) \(0.6735958333\)
\(L(1)\) \(\approx\) \(0.6735958333\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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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

−28.2777508593885251071243785124, −27.16993874013527127640598618869, −26.45062752223186410497249679651, −25.12722673997755810689426183462, −24.36885252665476993582446970674, −23.47551790983017242674810756164, −21.86309112577170801110917454303, −21.25411916531801302218051561726, −20.27165241067000773486691404328, −18.61708181289913397780151729029, −18.03689497336059992364287860350, −17.290846313386303574799015446661, −16.47194047808442585664902002840, −15.29572154160859200292278624101, −13.96363904664682103513657375584, −12.3805067879768467489193764880, −11.54492649622796222524456989200, −10.23048271069715653442523981072, −9.91156249873310403966564384953, −8.16645765241551769341714204579, −7.17147575839318910614767619462, −5.810214282540041755093374540385, −5.0087460832031986886102738159, −2.49610036499725476315941021430, −1.19943853986274091625122764852, 1.19943853986274091625122764852, 2.49610036499725476315941021430, 5.0087460832031986886102738159, 5.810214282540041755093374540385, 7.17147575839318910614767619462, 8.16645765241551769341714204579, 9.91156249873310403966564384953, 10.23048271069715653442523981072, 11.54492649622796222524456989200, 12.3805067879768467489193764880, 13.96363904664682103513657375584, 15.29572154160859200292278624101, 16.47194047808442585664902002840, 17.290846313386303574799015446661, 18.03689497336059992364287860350, 18.61708181289913397780151729029, 20.27165241067000773486691404328, 21.25411916531801302218051561726, 21.86309112577170801110917454303, 23.47551790983017242674810756164, 24.36885252665476993582446970674, 25.12722673997755810689426183462, 26.45062752223186410497249679651, 27.16993874013527127640598618869, 28.2777508593885251071243785124

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