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Properties

Label 2-847-77.62-c1-0-11
Degree $2$
Conductor $847$
Sign $-0.953 - 0.301i$
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  + (−0.513 + 0.707i)2-s + (−1.26 − 0.410i)3-s + (0.381 + 1.17i)4-s + (0.780 + 1.07i)5-s + (0.939 − 0.682i)6-s + (1.68 + 2.03i)7-s + (−2.68 − 0.874i)8-s + (−0.999 − 0.726i)9-s − 1.16·10-s − 1.64i·12-s + (4.91 + 3.57i)13-s + (−2.30 + 0.148i)14-s + (−0.545 − 1.67i)15-s + (−3.39 + 2.46i)17-s + (1.02 − 0.333i)18-s + (−0.939 + 2.89i)19-s + ⋯
L(s)  = 1  + (−0.363 + 0.499i)2-s + (−0.729 − 0.236i)3-s + (0.190 + 0.587i)4-s + (0.349 + 0.480i)5-s + (0.383 − 0.278i)6-s + (0.638 + 0.769i)7-s + (−0.951 − 0.309i)8-s + (−0.333 − 0.242i)9-s − 0.367·10-s − 0.473i·12-s + (1.36 + 0.990i)13-s + (−0.616 + 0.0396i)14-s + (−0.140 − 0.433i)15-s + (−0.824 + 0.598i)17-s + (0.242 − 0.0786i)18-s + (−0.215 + 0.663i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.953 - 0.301i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.953 - 0.301i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134572 + 0.872290i\)
\(L(\frac12)\) \(\approx\) \(0.134572 + 0.872290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.68 - 2.03i)T \)
11 \( 1 \)
good2 \( 1 + (0.513 - 0.707i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (1.26 + 0.410i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.780 - 1.07i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-4.91 - 3.57i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.39 - 2.46i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.939 - 2.89i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 + (6.52 - 2.12i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.30 + 4.55i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.927 + 2.85i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.939 - 2.89i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.61iT - 43T^{2} \)
47 \( 1 + (9.14 + 2.96i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.61 - 1.17i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (6.91 - 2.24i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.45 + 1.78i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 + (0.809 - 0.587i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.09 + 6.46i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.49 - 3.43i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.01 - 5.09i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 18.5iT - 89T^{2} \)
97 \( 1 + (-3.30 + 4.55i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.96842671407080454883186954674, −9.449527827617044562553668334708, −8.710773726833154168074729636739, −8.159655740250813819948566651952, −6.92989878034263683357053763752, −6.25283598199698488723488524221, −5.81628374430336708071702776405, −4.33558881181424311653931224390, −3.10202877857855113984194822896, −1.78176126754890103840823154653, 0.53675896219274803697470515961, 1.64264253898615858236938409466, 3.12004314419596630753861899821, 4.67643568579143134741505547540, 5.33821757553704159197992911569, 6.11881233538459432083559767778, 7.14822871349048394951460217984, 8.460780269930305778782711137931, 8.993696189945447382263699139182, 10.10971458124073352522123642215

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