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Properties

Label 2-804-201.38-c1-0-7
Degree $2$
Conductor $804$
Sign $0.923 + 0.383i$
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  + (−0.870 − 1.49i)3-s − 2.31·5-s + (−0.381 + 0.220i)7-s + (−1.48 + 2.60i)9-s + (1.81 + 3.13i)11-s + (1.86 + 1.07i)13-s + (2.01 + 3.46i)15-s + (3.58 + 2.07i)17-s + (2.79 − 4.83i)19-s + (0.662 + 0.379i)21-s + (−3.74 − 2.16i)23-s + 0.350·25-s + (5.19 − 0.0453i)27-s + (1.35 − 0.783i)29-s + (9.56 − 5.52i)31-s + ⋯
L(s)  = 1  + (−0.502 − 0.864i)3-s − 1.03·5-s + (−0.144 + 0.0832i)7-s + (−0.494 + 0.868i)9-s + (0.546 + 0.946i)11-s + (0.516 + 0.298i)13-s + (0.519 + 0.894i)15-s + (0.869 + 0.502i)17-s + (0.641 − 1.11i)19-s + (0.144 + 0.0828i)21-s + (−0.780 − 0.450i)23-s + 0.0700·25-s + (0.999 − 0.00872i)27-s + (0.251 − 0.145i)29-s + (1.71 − 0.991i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.923 + 0.383i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.923 + 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02846 - 0.204888i\)
\(L(\frac12)\) \(\approx\) \(1.02846 - 0.204888i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.870 + 1.49i)T \)
67 \( 1 + (-7.39 + 3.51i)T \)
good5 \( 1 + 2.31T + 5T^{2} \)
7 \( 1 + (0.381 - 0.220i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.81 - 3.13i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.86 - 1.07i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.58 - 2.07i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.79 + 4.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.74 + 2.16i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.35 + 0.783i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-9.56 + 5.52i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.08 + 3.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.96 + 6.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 10.1iT - 43T^{2} \)
47 \( 1 + (-1.58 + 0.914i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.10T + 53T^{2} \)
59 \( 1 - 11.2iT - 59T^{2} \)
61 \( 1 + (-4.53 - 2.61i)T + (30.5 + 52.8i)T^{2} \)
71 \( 1 + (-8.03 + 4.64i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.45 - 5.98i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.57 + 5.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-15.2 - 8.82i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.87iT - 89T^{2} \)
97 \( 1 + (-3.26 - 1.88i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.32287012965075333693915222926, −9.328803521669119897392694951276, −8.184465456680260060458802977899, −7.66079991197184589035736314753, −6.76986625910442583635087274656, −6.02484649605761938973004022006, −4.77639377189672084203408020781, −3.86226904249358288606773959321, −2.42008614844323256582647596272, −0.917161921221095707844425432608, 0.840859626709564374149313565442, 3.40331116281434800837229854575, 3.64334337997038007805106382353, 4.91652777913004049266594723661, 5.83235098821356572028290672034, 6.69736017131830244353359366231, 8.018329466699530921386631250446, 8.472302371789609336529483902604, 9.712429022309499056040548408084, 10.23836077519473592800225398253

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