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Properties

Label 2-804-268.7-c1-0-54
Degree $2$
Conductor $804$
Sign $-0.257 + 0.966i$
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  + (0.319 − 1.37i)2-s + (0.959 + 0.281i)3-s + (−1.79 − 0.880i)4-s + (2.42 + 2.09i)5-s + (0.694 − 1.23i)6-s + (0.217 − 4.57i)7-s + (−1.78 + 2.19i)8-s + (0.841 + 0.540i)9-s + (3.66 − 2.66i)10-s + (−4.38 − 0.845i)11-s + (−1.47 − 1.35i)12-s + (2.32 − 5.81i)13-s + (−6.23 − 1.76i)14-s + (1.73 + 2.69i)15-s + (2.44 + 3.16i)16-s + (1.76 + 2.48i)17-s + ⋯
L(s)  = 1  + (0.226 − 0.974i)2-s + (0.553 + 0.162i)3-s + (−0.897 − 0.440i)4-s + (1.08 + 0.938i)5-s + (0.283 − 0.502i)6-s + (0.0823 − 1.72i)7-s + (−0.631 + 0.775i)8-s + (0.280 + 0.180i)9-s + (1.15 − 0.842i)10-s + (−1.32 − 0.255i)11-s + (−0.425 − 0.389i)12-s + (0.645 − 1.61i)13-s + (−1.66 − 0.471i)14-s + (0.447 + 0.695i)15-s + (0.612 + 0.790i)16-s + (0.428 + 0.602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.257 + 0.966i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.257 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30983 - 1.70467i\)
\(L(\frac12)\) \(\approx\) \(1.30983 - 1.70467i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.319 + 1.37i)T \)
3 \( 1 + (-0.959 - 0.281i)T \)
67 \( 1 + (-8.11 - 1.08i)T \)
good5 \( 1 + (-2.42 - 2.09i)T + (0.711 + 4.94i)T^{2} \)
7 \( 1 + (-0.217 + 4.57i)T + (-6.96 - 0.665i)T^{2} \)
11 \( 1 + (4.38 + 0.845i)T + (10.2 + 4.08i)T^{2} \)
13 \( 1 + (-2.32 + 5.81i)T + (-9.40 - 8.97i)T^{2} \)
17 \( 1 + (-1.76 - 2.48i)T + (-5.56 + 16.0i)T^{2} \)
19 \( 1 + (-1.08 + 0.0514i)T + (18.9 - 1.80i)T^{2} \)
23 \( 1 + (1.85 + 1.94i)T + (-1.09 + 22.9i)T^{2} \)
29 \( 1 + (0.373 + 0.646i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.46 + 3.38i)T + (22.4 - 21.3i)T^{2} \)
37 \( 1 + (-4.72 + 8.18i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.787 - 8.24i)T + (-40.2 + 7.75i)T^{2} \)
43 \( 1 + (2.54 - 5.57i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + (2.93 - 0.712i)T + (41.7 - 21.5i)T^{2} \)
53 \( 1 + (1.61 - 0.738i)T + (34.7 - 40.0i)T^{2} \)
59 \( 1 + (-2.20 - 0.317i)T + (56.6 + 16.6i)T^{2} \)
61 \( 1 + (-1.18 - 6.17i)T + (-56.6 + 22.6i)T^{2} \)
71 \( 1 + (3.14 + 2.23i)T + (23.2 + 67.0i)T^{2} \)
73 \( 1 + (-0.313 + 0.0605i)T + (67.7 - 27.1i)T^{2} \)
79 \( 1 + (7.97 - 6.27i)T + (18.6 - 76.7i)T^{2} \)
83 \( 1 + (5.38 - 1.86i)T + (65.2 - 51.3i)T^{2} \)
89 \( 1 + (7.32 - 2.14i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-2.04 - 1.18i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.11963057321082009503616860997, −9.820038248772569477453541572822, −8.179485800825954332642336886551, −7.78182035089783039078468232434, −6.32987001775918757978974237499, −5.47906486801937048571461946464, −4.25519960281581529771387279512, −3.22357646107577244278029856319, −2.56322951679980655199627564105, −1.00819466096981725897409298715, 1.80971933614856603740532142620, 2.92078092701742200553490730678, 4.60651521840761664852062011288, 5.32900998637695215668935270494, 5.97140343969880267196284039241, 6.95403726807674564397691850156, 8.288200397611956270192297126808, 8.591759065178565080319696907826, 9.457178018498018021065819666145, 9.861600381625656402679652400923

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