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Properties

Label 2-804-12.11-c1-0-105
Degree $2$
Conductor $804$
Sign $-0.193 + 0.981i$
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  + (1.41 − 0.108i)2-s + (−1.62 − 0.591i)3-s + (1.97 − 0.307i)4-s − 3.04i·5-s + (−2.35 − 0.656i)6-s − 0.278i·7-s + (2.75 − 0.648i)8-s + (2.30 + 1.92i)9-s + (−0.332 − 4.29i)10-s + 5.07·11-s + (−3.39 − 0.669i)12-s − 5.86·13-s + (−0.0303 − 0.392i)14-s + (−1.80 + 4.96i)15-s + (3.81 − 1.21i)16-s − 7.60i·17-s + ⋯
L(s)  = 1  + (0.997 − 0.0769i)2-s + (−0.939 − 0.341i)3-s + (0.988 − 0.153i)4-s − 1.36i·5-s + (−0.963 − 0.268i)6-s − 0.105i·7-s + (0.973 − 0.229i)8-s + (0.766 + 0.642i)9-s + (−0.104 − 1.35i)10-s + 1.53·11-s + (−0.981 − 0.193i)12-s − 1.62·13-s + (−0.00810 − 0.105i)14-s + (−0.465 + 1.28i)15-s + (0.952 − 0.303i)16-s − 1.84i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.193 + 0.981i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.193 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37171 - 1.66818i\)
\(L(\frac12)\) \(\approx\) \(1.37171 - 1.66818i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.108i)T \)
3 \( 1 + (1.62 + 0.591i)T \)
67 \( 1 - iT \)
good5 \( 1 + 3.04iT - 5T^{2} \)
7 \( 1 + 0.278iT - 7T^{2} \)
11 \( 1 - 5.07T + 11T^{2} \)
13 \( 1 + 5.86T + 13T^{2} \)
17 \( 1 + 7.60iT - 17T^{2} \)
19 \( 1 - 5.91iT - 19T^{2} \)
23 \( 1 + 2.24T + 23T^{2} \)
29 \( 1 + 5.83iT - 29T^{2} \)
31 \( 1 + 4.34iT - 31T^{2} \)
37 \( 1 + 6.86T + 37T^{2} \)
41 \( 1 - 8.48iT - 41T^{2} \)
43 \( 1 - 2.79iT - 43T^{2} \)
47 \( 1 + 0.455T + 47T^{2} \)
53 \( 1 + 5.43iT - 53T^{2} \)
59 \( 1 - 0.908T + 59T^{2} \)
61 \( 1 - 1.16T + 61T^{2} \)
71 \( 1 - 4.54T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 - 12.3iT - 79T^{2} \)
83 \( 1 + 2.79T + 83T^{2} \)
89 \( 1 - 1.69iT - 89T^{2} \)
97 \( 1 - 4.26T + 97T^{2} \)
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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

−9.933691024665045940457741805400, −9.540393428107385293829124063985, −8.009991040760015699880051149737, −7.20053873279332014458304306882, −6.33436899698575241903966077210, −5.35741529642030387593353174365, −4.75913086742199186392526744541, −3.97581428192111826302082297431, −2.13358579404964503360555721188, −0.903244089947755087528996990359, 1.94836165780551283101616156590, 3.33249303383257289597550645835, 4.14707098479136679750050515524, 5.17327345921095722264015850585, 6.16416786960567240665816169343, 6.88784379965435374832667964194, 7.21517836427880022256994670953, 8.920575680076430846861977170760, 10.14505927262750507116152576908, 10.65959316825531961171839526578

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