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Properties

Label 2-804-67.37-c1-0-7
Degree $2$
Conductor $804$
Sign $0.747 + 0.663i$
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 + 3.38·5-s + (1.69 − 2.93i)7-s + 9-s + (1 − 1.73i)11-s + (1.53 + 2.65i)13-s − 3.38·15-s + (−1.03 − 1.78i)17-s + (−3.72 − 6.45i)19-s + (−1.69 + 2.93i)21-s + (1.11 + 1.92i)23-s + 6.44·25-s − 27-s + (−2.80 + 4.85i)29-s + (−2.99 + 5.18i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·5-s + (0.639 − 1.10i)7-s + 0.333·9-s + (0.301 − 0.522i)11-s + (0.425 + 0.736i)13-s − 0.873·15-s + (−0.250 − 0.433i)17-s + (−0.854 − 1.47i)19-s + (−0.369 + 0.639i)21-s + (0.231 + 0.401i)23-s + 1.28·25-s − 0.192·27-s + (−0.520 + 0.901i)29-s + (−0.537 + 0.931i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.747 + 0.663i$
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: \((2,\ 804,\ (\ :1/2),\ 0.747 + 0.663i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67553 - 0.636258i\)
\(L(\frac12)\) \(\approx\) \(1.67553 - 0.636258i\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.80 + 2.46i)T \)
good5 \( 1 - 3.38T + 5T^{2} \)
7 \( 1 + (-1.69 + 2.93i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.53 - 2.65i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.03 + 1.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.72 + 6.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.11 - 1.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.80 - 4.85i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.99 - 5.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.14 + 5.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.92 + 3.32i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 5.92T + 43T^{2} \)
47 \( 1 + (-1.92 + 3.32i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.93T + 53T^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 + (-4.98 - 8.63i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-6.65 + 11.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.80 - 4.85i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.29 - 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.03 + 3.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.837T + 89T^{2} \)
97 \( 1 + (-3.08 - 5.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.40875435245283119179744374133, −9.223471375414101183192512448697, −8.815634569396388427500147974383, −7.19231452048336106549494591254, −6.75812415467327417917281660210, −5.70628928602235235954802316651, −4.92870265848054464743164173312, −3.88209008676129061588437495665, −2.20601558774055147003288484817, −1.07543074553081861579739367971, 1.61724318612866070566807204850, 2.38898790673550169901712074357, 4.14012478527300088394886741299, 5.38318519546251203009101496077, 5.84707238469448003540157817085, 6.50472597271883575971069106001, 7.925343345848199609060117330142, 8.739138674499467502975916768821, 9.655083046063271155137123561018, 10.27984006414596316050924036735

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