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Properties

Label 2-806-403.315-c1-0-30
Degree $2$
Conductor $806$
Sign $0.336 + 0.941i$
Analytic cond. $6.43594$
Root an. cond. $2.53691$
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  + (0.5 + 0.866i)2-s − 0.369·3-s + (−0.499 + 0.866i)4-s + (0.420 + 0.729i)5-s + (−0.184 − 0.319i)6-s + (−1.03 − 1.79i)7-s − 0.999·8-s − 2.86·9-s + (−0.420 + 0.729i)10-s + (2.76 − 4.78i)11-s + (0.184 − 0.319i)12-s + (−3.50 − 0.851i)13-s + (1.03 − 1.79i)14-s + (−0.155 − 0.269i)15-s + (−0.5 − 0.866i)16-s + (−3.19 − 5.54i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s − 0.213·3-s + (−0.249 + 0.433i)4-s + (0.188 + 0.326i)5-s + (−0.0753 − 0.130i)6-s + (−0.391 − 0.677i)7-s − 0.353·8-s − 0.954·9-s + (−0.133 + 0.230i)10-s + (0.832 − 1.44i)11-s + (0.0532 − 0.0922i)12-s + (−0.971 − 0.236i)13-s + (0.276 − 0.479i)14-s + (−0.0401 − 0.0694i)15-s + (−0.125 − 0.216i)16-s + (−0.776 − 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 806 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 806 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(806\)    =    \(2 \cdot 13 \cdot 31\)
Sign: $0.336 + 0.941i$
Analytic conductor: \(6.43594\)
Root analytic conductor: \(2.53691\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{806} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 806,\ (\ :1/2),\ 0.336 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.756761 - 0.533189i\)
\(L(\frac12)\) \(\approx\) \(0.756761 - 0.533189i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (3.50 + 0.851i)T \)
31 \( 1 + (1.81 - 5.26i)T \)
good3 \( 1 + 0.369T + 3T^{2} \)
5 \( 1 + (-0.420 - 0.729i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.03 + 1.79i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.76 + 4.78i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.19 + 5.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.166 + 0.288i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.147 - 0.255i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.32 - 7.48i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + 8.27T + 37T^{2} \)
41 \( 1 + (-3.43 + 5.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.01 + 8.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.37T + 47T^{2} \)
53 \( 1 + (4.52 + 7.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.47 - 7.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.977 - 1.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.75 + 6.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + (-6.54 - 11.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.58 - 7.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.62 + 4.54i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.52 + 2.64i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.04 + 10.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.17642879098055492499610251324, −8.894049528996543963292764665396, −8.585192644660059966794204594163, −6.98839699659581155969010276138, −6.85978646226319936487636131001, −5.66763782904318783920296887779, −4.92600696814074537650371465838, −3.57518379849573952667548037796, −2.78048695511085415033491165925, −0.40119775645709006066080770700, 1.78699632096556640391962624382, 2.72397657942225009855783619905, 4.14346936910066283610645595351, 4.92024399288704079610749953290, 5.98497154563602532710208411723, 6.65914659445318995399985739938, 7.997518177337864966421699108195, 9.096159483442544647216103449807, 9.511074274894699315374332831233, 10.43388087882503209885565785151

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