Properties

Label 2-451-451.119-c1-0-3
Degree $2$
Conductor $451$
Sign $0.984 + 0.175i$
Analytic cond. $3.60125$
Root an. cond. $1.89769$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.804 − 2.47i)2-s + (0.229 − 0.166i)3-s + (−3.86 + 2.80i)4-s − 1.16·5-s + (−0.596 − 0.433i)6-s + (−0.525 − 1.61i)7-s + (5.85 + 4.25i)8-s + (−0.902 + 2.77i)9-s + (0.941 + 2.89i)10-s + (2.26 + 2.42i)11-s + (−0.418 + 1.28i)12-s + (1.00 + 3.08i)13-s + (−3.58 + 2.60i)14-s + (−0.267 + 0.194i)15-s + (2.86 − 8.81i)16-s + (−6.43 + 4.67i)17-s + ⋯
L(s)  = 1  + (−0.568 − 1.75i)2-s + (0.132 − 0.0960i)3-s + (−1.93 + 1.40i)4-s − 0.523·5-s + (−0.243 − 0.176i)6-s + (−0.198 − 0.611i)7-s + (2.06 + 1.50i)8-s + (−0.300 + 0.925i)9-s + (0.297 + 0.915i)10-s + (0.683 + 0.730i)11-s + (−0.120 + 0.371i)12-s + (0.278 + 0.856i)13-s + (−0.957 + 0.695i)14-s + (−0.0691 + 0.0502i)15-s + (0.716 − 2.20i)16-s + (−1.56 + 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(451\)    =    \(11 \cdot 41\)
Sign: $0.984 + 0.175i$
Analytic conductor: \(3.60125\)
Root analytic conductor: \(1.89769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{451} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 451,\ (\ :1/2),\ 0.984 + 0.175i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.572696 - 0.0505721i\)
\(L(\frac12)\) \(\approx\) \(0.572696 - 0.0505721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-2.26 - 2.42i)T \)
41 \( 1 + (0.000300 - 6.40i)T \)
good2 \( 1 + (0.804 + 2.47i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.229 + 0.166i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + 1.16T + 5T^{2} \)
7 \( 1 + (0.525 + 1.61i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.00 - 3.08i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (6.43 - 4.67i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + 0.642T + 19T^{2} \)
23 \( 1 + (-7.42 + 5.39i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (5.26 - 3.82i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 - 4.54T + 31T^{2} \)
37 \( 1 + (-7.97 + 5.79i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (0.459 - 0.334i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (3.41 - 10.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.91 + 4.29i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + 7.36T + 59T^{2} \)
61 \( 1 + (3.18 - 9.79i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (3.14 - 9.68i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (5.12 - 3.72i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-7.38 - 5.36i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.33 - 4.10i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.18 - 6.73i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (7.43 + 5.40i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-12.1 - 8.85i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.10937054385435889714200817005, −10.44674226948582585642404968034, −9.338049125776097903579002572802, −8.722258946903771872025582516668, −7.78204824782912598714782237818, −6.64126431765513692408287835973, −4.47781602661990780621196313215, −4.10981967927298364359376740603, −2.64833933969607813221730672060, −1.53359827781495472349603505729, 0.45322365550991947372457673788, 3.33103542953429442528319784598, 4.68792977433084188435284996972, 5.84115666347562426193124385563, 6.46315037171752346071658483294, 7.43667367740876214943877822056, 8.364886277114328134981884410570, 9.165791779793964740976093203340, 9.470528476306157856232335381871, 11.06846925119390634984008462248

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