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Properties

Label 2-151-151.150-c0-0-0
Degree $2$
Conductor $151$
Sign $1$
Analytic cond. $0.0753588$
Root an. cond. $0.274515$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.80·2-s + 2.24·4-s − 0.445·5-s − 2.24·8-s + 9-s + 0.801·10-s + 1.24·11-s + 1.80·16-s + 1.24·17-s − 1.80·18-s − 1.80·19-s − 20-s − 2.24·22-s − 0.801·25-s − 1.80·29-s − 0.445·31-s − 1.00·32-s − 2.24·34-s + 2.24·36-s + 1.24·37-s + 3.24·38-s + 1.00·40-s − 0.445·43-s + 2.80·44-s − 0.445·45-s − 1.80·47-s + 49-s + ⋯
L(s)  = 1  − 1.80·2-s + 2.24·4-s − 0.445·5-s − 2.24·8-s + 9-s + 0.801·10-s + 1.24·11-s + 1.80·16-s + 1.24·17-s − 1.80·18-s − 1.80·19-s − 20-s − 2.24·22-s − 0.801·25-s − 1.80·29-s − 0.445·31-s − 1.00·32-s − 2.24·34-s + 2.24·36-s + 1.24·37-s + 3.24·38-s + 1.00·40-s − 0.445·43-s + 2.80·44-s − 0.445·45-s − 1.80·47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(151\)
Sign: $1$
Analytic conductor: \(0.0753588\)
Root analytic conductor: \(0.274515\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (150, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 151,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3183807659\)
\(L(\frac12)\) \(\approx\) \(0.3183807659\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 - T \)
good2 \( 1 + 1.80T + T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 + 0.445T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.24T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 1.24T + T^{2} \)
19 \( 1 + 1.80T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.80T + T^{2} \)
31 \( 1 + 0.445T + T^{2} \)
37 \( 1 - 1.24T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 0.445T + T^{2} \)
47 \( 1 + 1.80T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 0.445T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.80T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.90151542292096618877600356773, −11.85438620780584819009386000933, −10.99704340753625651184858104778, −9.949700238281880146439100369668, −9.243627037633449600472867243059, −8.116820828412582764542650046626, −7.27784716890115809412202436983, −6.24830716053211365215499522053, −3.92589233723069360045659193864, −1.69473509693998334125644510580, 1.69473509693998334125644510580, 3.92589233723069360045659193864, 6.24830716053211365215499522053, 7.27784716890115809412202436983, 8.116820828412582764542650046626, 9.243627037633449600472867243059, 9.949700238281880146439100369668, 10.99704340753625651184858104778, 11.85438620780584819009386000933, 12.90151542292096618877600356773

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