Properties

Label 2-804-201.5-c1-0-7
Degree $2$
Conductor $804$
Sign $0.582 + 0.812i$
Analytic cond. $6.41997$
Root an. cond. $2.53376$
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  + (−1.53 − 0.807i)3-s + (−3.45 + 1.01i)5-s + (−0.403 − 0.349i)7-s + (1.69 + 2.47i)9-s + (0.843 − 0.247i)11-s + (2.49 + 3.88i)13-s + (6.10 + 1.23i)15-s + (−6.49 − 0.934i)17-s + (−1.01 − 1.17i)19-s + (0.335 + 0.860i)21-s + (1.87 − 0.854i)23-s + (6.68 − 4.29i)25-s + (−0.601 − 5.16i)27-s − 3.54i·29-s + (2.94 − 4.58i)31-s + ⋯
L(s)  = 1  + (−0.884 − 0.466i)3-s + (−1.54 + 0.453i)5-s + (−0.152 − 0.132i)7-s + (0.565 + 0.824i)9-s + (0.254 − 0.0746i)11-s + (0.692 + 1.07i)13-s + (1.57 + 0.318i)15-s + (−1.57 − 0.226i)17-s + (−0.233 − 0.269i)19-s + (0.0732 + 0.187i)21-s + (0.390 − 0.178i)23-s + (1.33 − 0.859i)25-s + (−0.115 − 0.993i)27-s − 0.658i·29-s + (0.529 − 0.824i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.582 + 0.812i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.582 + 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.575971 - 0.295629i\)
\(L(\frac12)\) \(\approx\) \(0.575971 - 0.295629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.53 + 0.807i)T \)
67 \( 1 + (-4.80 - 6.62i)T \)
good5 \( 1 + (3.45 - 1.01i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (0.403 + 0.349i)T + (0.996 + 6.92i)T^{2} \)
11 \( 1 + (-0.843 + 0.247i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-2.49 - 3.88i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (6.49 + 0.934i)T + (16.3 + 4.78i)T^{2} \)
19 \( 1 + (1.01 + 1.17i)T + (-2.70 + 18.8i)T^{2} \)
23 \( 1 + (-1.87 + 0.854i)T + (15.0 - 17.3i)T^{2} \)
29 \( 1 + 3.54iT - 29T^{2} \)
31 \( 1 + (-2.94 + 4.58i)T + (-12.8 - 28.1i)T^{2} \)
37 \( 1 - 6.30T + 37T^{2} \)
41 \( 1 + (-1.58 + 11.0i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-4.21 - 0.605i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (-8.24 + 3.76i)T + (30.7 - 35.5i)T^{2} \)
53 \( 1 + (-0.880 - 6.12i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (0.431 - 0.671i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (1.65 - 5.64i)T + (-51.3 - 32.9i)T^{2} \)
71 \( 1 + (-3.24 + 0.465i)T + (68.1 - 20.0i)T^{2} \)
73 \( 1 + (4.50 + 1.32i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (1.15 + 1.79i)T + (-32.8 + 71.8i)T^{2} \)
83 \( 1 + (-0.393 - 1.33i)T + (-69.8 + 44.8i)T^{2} \)
89 \( 1 + (0.163 + 0.0744i)T + (58.2 + 67.2i)T^{2} \)
97 \( 1 + 3.07iT - 97T^{2} \)
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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

−10.53760374820749730474195417600, −9.156512706454600240339929443174, −8.319587611217585206977362052900, −7.25985673934216251520333096749, −6.84459704519679506086099274447, −5.94632988975914509560231668607, −4.38721071812729011279801569972, −4.07181784309376991622818051616, −2.37261133895072922837406926732, −0.51995409707330855804189510160, 0.898418380090429982895599451510, 3.21584556902435003636067472856, 4.18985701019165991686612289651, 4.80579059028334093697480007681, 5.99390845880040466625927234045, 6.84464451370913333134906402820, 7.905015834789773684031886330769, 8.657948098148145213501886961739, 9.517107570168215812157607283430, 10.85936296864479860180308961487

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