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Properties

Label 2-847-11.9-c1-0-35
Degree $2$
Conductor $847$
Sign $0.923 - 0.382i$
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  + (−0.835 + 2.57i)2-s + (2.27 − 1.65i)3-s + (−4.29 − 3.11i)4-s + (−0.137 − 0.423i)5-s + (2.34 + 7.21i)6-s + (0.809 + 0.587i)7-s + (7.23 − 5.25i)8-s + (1.50 − 4.64i)9-s + 1.20·10-s − 14.8·12-s + (−0.139 + 0.428i)13-s + (−2.18 + 1.58i)14-s + (−1.01 − 0.734i)15-s + (4.18 + 12.8i)16-s + (−1.49 − 4.59i)17-s + (10.6 + 7.75i)18-s + ⋯
L(s)  = 1  + (−0.590 + 1.81i)2-s + (1.31 − 0.952i)3-s + (−2.14 − 1.55i)4-s + (−0.0615 − 0.189i)5-s + (0.957 + 2.94i)6-s + (0.305 + 0.222i)7-s + (2.55 − 1.85i)8-s + (0.502 − 1.54i)9-s + 0.380·10-s − 4.30·12-s + (−0.0386 + 0.118i)13-s + (−0.584 + 0.424i)14-s + (−0.260 − 0.189i)15-s + (1.04 + 3.22i)16-s + (−0.362 − 1.11i)17-s + (2.51 + 1.82i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (372, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52316 + 0.302912i\)
\(L(\frac12)\) \(\approx\) \(1.52316 + 0.302912i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.835 - 2.57i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-2.27 + 1.65i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.137 + 0.423i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.139 - 0.428i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.49 + 4.59i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.878 - 0.638i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.57T + 23T^{2} \)
29 \( 1 + (1.60 + 1.16i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.55 + 7.85i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.91 + 4.30i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.43 + 1.04i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + (0.827 - 0.601i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.10 - 3.39i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-11.6 - 8.46i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.52 + 4.68i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 6.18T + 67T^{2} \)
71 \( 1 + (1.82 + 5.63i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.34 + 0.974i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.11 + 3.43i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.33 - 10.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 5.21T + 89T^{2} \)
97 \( 1 + (1.64 - 5.04i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.435181155723760097874483901798, −9.067239466780725852419820479018, −8.385774303145038851189867948217, −7.61999227538996292817035229968, −7.14775356469105089334167456141, −6.30411317416083776032893079540, −5.22132748647412571248868998375, −4.16232467223599842832902985890, −2.47007840246440174567259289915, −0.894340575284045330360530106090, 1.51436698318893673174203371048, 2.68895519090731991360609253892, 3.38662990713589809931648116990, 4.18908636257058571581414158225, 5.00594734935092021517065560222, 7.19948067220437676205940276117, 8.362858895883134978098177778677, 8.668684092412867538845018996911, 9.401580474708156331732544171083, 10.27978976460735466285890238608

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