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Properties

Label 2-804-67.37-c1-0-4
Degree $2$
Conductor $804$
Sign $0.144 + 0.989i$
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  − 3-s − 2.60·5-s + (−1.30 + 2.25i)7-s + 9-s + (1 − 1.73i)11-s + (2.19 + 3.79i)13-s + 2.60·15-s + (−1.69 − 2.92i)17-s + (−1.38 − 2.40i)19-s + (1.30 − 2.25i)21-s + (−0.798 − 1.38i)23-s + 1.77·25-s − 27-s + (2.10 − 3.63i)29-s + (4.90 − 8.49i)31-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.16·5-s + (−0.492 + 0.852i)7-s + 0.333·9-s + (0.301 − 0.522i)11-s + (0.607 + 1.05i)13-s + 0.672·15-s + (−0.410 − 0.710i)17-s + (−0.318 − 0.552i)19-s + (0.284 − 0.492i)21-s + (−0.166 − 0.288i)23-s + 0.355·25-s − 0.192·27-s + (0.390 − 0.675i)29-s + (0.880 − 1.52i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.144 + 0.989i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.144 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.458105 - 0.395901i\)
\(L(\frac12)\) \(\approx\) \(0.458105 - 0.395901i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
67 \( 1 + (4.38 + 6.91i)T \)
good5 \( 1 + 2.60T + 5T^{2} \)
7 \( 1 + (1.30 - 2.25i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.19 - 3.79i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.69 + 2.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.38 + 2.40i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.798 + 1.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.10 + 3.63i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.90 + 8.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.89 + 3.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.48 + 7.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + (-4.48 + 7.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.618T + 53T^{2} \)
59 \( 1 - 3.18T + 59T^{2} \)
61 \( 1 + (-7.38 - 12.7i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (3.40 - 5.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.613 + 1.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.01 + 8.68i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.69 + 4.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.00T + 89T^{2} \)
97 \( 1 + (-1.99 - 3.45i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.11502223638349467026519223454, −9.042216986103770434895224750959, −8.538549360898822213656019113148, −7.37605835653638623898590917850, −6.53911488374367584507598632603, −5.77818393723205166091892800997, −4.50787884572904726950799130310, −3.77762922228934395036255792250, −2.39741867783611540227180623641, −0.37889026753825062223188651647, 1.17785955714102836814123734224, 3.29562754982060815426501113775, 4.02460548060863016419149939512, 4.96431211210260595867401030721, 6.27592349812336339839264744024, 6.92870829682754075740543764410, 7.903109571427510777610467504156, 8.514520239935621510090049179692, 9.917334350112445055836603883028, 10.49199242232514090695812462998

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