Properties

Label 2-847-77.4-c1-0-56
Degree $2$
Conductor $847$
Sign $-0.508 + 0.861i$
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  + (0.0686 − 0.653i)2-s + (1.27 + 1.42i)3-s + (1.53 + 0.326i)4-s + (−3.26 − 1.45i)5-s + (1.01 − 0.738i)6-s + (−2.40 − 1.09i)7-s + (0.724 − 2.22i)8-s + (−0.0686 + 0.653i)9-s + (−1.17 + 2.02i)10-s + (1.5 + 2.59i)12-s + (−4.78 − 3.47i)13-s + (−0.879 + 1.49i)14-s + (−2.10 − 6.49i)15-s + (1.46 + 0.650i)16-s + (−0.173 − 1.64i)17-s + (0.421 + 0.0896i)18-s + ⋯
L(s)  = 1  + (0.0485 − 0.461i)2-s + (0.738 + 0.820i)3-s + (0.767 + 0.163i)4-s + (−1.45 − 0.649i)5-s + (0.414 − 0.301i)6-s + (−0.910 − 0.413i)7-s + (0.256 − 0.787i)8-s + (−0.0228 + 0.217i)9-s + (−0.370 + 0.641i)10-s + (0.433 + 0.749i)12-s + (−1.32 − 0.963i)13-s + (−0.235 + 0.400i)14-s + (−0.544 − 1.67i)15-s + (0.365 + 0.162i)16-s + (−0.0419 − 0.399i)17-s + (0.0994 + 0.0211i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.508 + 0.861i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.508 + 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.575895 - 1.00887i\)
\(L(\frac12)\) \(\approx\) \(0.575895 - 1.00887i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.40 + 1.09i)T \)
11 \( 1 \)
good2 \( 1 + (-0.0686 + 0.653i)T + (-1.95 - 0.415i)T^{2} \)
3 \( 1 + (-1.27 - 1.42i)T + (-0.313 + 2.98i)T^{2} \)
5 \( 1 + (3.26 + 1.45i)T + (3.34 + 3.71i)T^{2} \)
13 \( 1 + (4.78 + 3.47i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.173 + 1.64i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (-1.44 + 0.307i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (1.67 + 2.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.951 + 2.92i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.46 - 2.87i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (3.01 - 3.35i)T + (-3.86 - 36.7i)T^{2} \)
41 \( 1 + (-0.397 + 1.22i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.59T + 43T^{2} \)
47 \( 1 + (-1.62 + 0.344i)T + (42.9 - 19.1i)T^{2} \)
53 \( 1 + (-8.42 + 3.75i)T + (35.4 - 39.3i)T^{2} \)
59 \( 1 + (8.65 + 1.84i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + (-6.10 - 2.71i)T + (40.8 + 45.3i)T^{2} \)
67 \( 1 + (-4.91 + 8.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.97 + 5.06i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.46 - 0.948i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (0.668 - 6.35i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (-0.135 + 0.0986i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.28 - 2.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.87 + 5.72i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.967484197928675358261718335338, −9.232489336161546241842248792200, −8.177230695422003914351474130601, −7.50139861756659341482385711434, −6.73045199092899701221766775109, −5.12888817937985222051103210770, −4.04610592416039114009319048737, −3.46826208861896158032913123955, −2.65874392137752348941418219676, −0.46783707199136722260994173490, 2.03329333367379665593825168273, 2.88830406562608099447707713457, 3.88461892320387088923488891897, 5.38304609074527439504589349067, 6.61615737183595309512046413411, 7.23160408339741433776962416759, 7.54136844287800043658381186189, 8.461612570579504380860254660106, 9.479032202944544629919703145819, 10.58255339268716080310989097633

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