Properties

Label 2-847-77.37-c1-0-61
Degree $2$
Conductor $847$
Sign $-0.0959 + 0.995i$
Analytic cond. $6.76332$
Root an. cond. $2.60064$
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  + (2.27 − 1.01i)2-s + (0.697 − 0.148i)3-s + (2.81 − 3.12i)4-s + (−0.230 − 2.19i)5-s + (1.43 − 1.04i)6-s + (2.62 + 0.304i)7-s + (1.69 − 5.22i)8-s + (−2.27 + 1.01i)9-s + (−2.74 − 4.75i)10-s + (1.49 − 2.59i)12-s + (−2.65 − 1.93i)13-s + (6.28 − 1.96i)14-s + (−0.486 − 1.49i)15-s + (−0.550 − 5.23i)16-s + (−1.36 − 0.606i)17-s + (−4.15 + 4.61i)18-s + ⋯
L(s)  = 1  + (1.60 − 0.716i)2-s + (0.402 − 0.0856i)3-s + (1.40 − 1.56i)4-s + (−0.103 − 0.980i)5-s + (0.587 − 0.426i)6-s + (0.993 + 0.115i)7-s + (0.599 − 1.84i)8-s + (−0.758 + 0.337i)9-s + (−0.868 − 1.50i)10-s + (0.433 − 0.750i)12-s + (−0.737 − 0.535i)13-s + (1.68 − 0.526i)14-s + (−0.125 − 0.386i)15-s + (−0.137 − 1.30i)16-s + (−0.330 − 0.147i)17-s + (−0.978 + 1.08i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0959 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0959 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.0959 + 0.995i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (807, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.0959 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.93676 - 3.23357i\)
\(L(\frac12)\) \(\approx\) \(2.93676 - 3.23357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.62 - 0.304i)T \)
11 \( 1 \)
good2 \( 1 + (-2.27 + 1.01i)T + (1.33 - 1.48i)T^{2} \)
3 \( 1 + (-0.697 + 0.148i)T + (2.74 - 1.22i)T^{2} \)
5 \( 1 + (0.230 + 2.19i)T + (-4.89 + 1.03i)T^{2} \)
13 \( 1 + (2.65 + 1.93i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.36 + 0.606i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (-4.62 - 5.14i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (3.24 - 5.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.509 - 1.56i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.245 + 2.33i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-5.43 - 1.15i)T + (33.8 + 15.0i)T^{2} \)
41 \( 1 + (-3.47 + 10.6i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.26T + 43T^{2} \)
47 \( 1 + (-0.997 - 1.10i)T + (-4.91 + 46.7i)T^{2} \)
53 \( 1 + (0.0318 - 0.303i)T + (-51.8 - 11.0i)T^{2} \)
59 \( 1 + (8.47 - 9.40i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (1.35 + 12.9i)T + (-59.6 + 12.6i)T^{2} \)
67 \( 1 + (-2.28 - 3.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.16 - 6.65i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.73 + 6.36i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (4.23 - 1.88i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (1.56 - 1.13i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.60 - 2.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.50 + 1.09i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.24032668318515154526813633767, −9.196037626351996780915664457506, −8.154087325654529915946714052270, −7.49479221908567732173914106442, −5.79542715733358167038242686702, −5.34845507572151878316183975204, −4.59727257127134350594070920612, −3.57729500024209284905152328802, −2.48059946411942399379711747060, −1.45589842933598837655336580835, 2.47050225647113482224571834598, 3.08337323885610852576014606595, 4.30262234777249473284548015748, 4.94133021333295607328226078601, 6.06329470281150666160041813915, 6.80810335097268842563049813705, 7.55019733943879043076685883176, 8.346220616383841068077567311359, 9.497290234124008620787286098570, 10.79608146108852516962173577027

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