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Properties

Label 2-847-11.3-c1-0-50
Degree $2$
Conductor $847$
Sign $-0.995 - 0.0913i$
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  + (2.06 − 1.49i)2-s + (−0.126 − 0.390i)3-s + (1.39 − 4.29i)4-s + (−2.77 − 2.01i)5-s + (−0.847 − 0.615i)6-s + (−0.309 + 0.951i)7-s + (−1.98 − 6.09i)8-s + (2.29 − 1.66i)9-s − 8.75·10-s − 1.85·12-s + (−1.75 + 1.27i)13-s + (0.788 + 2.42i)14-s + (−0.435 + 1.33i)15-s + (−5.93 − 4.31i)16-s + (−3.65 − 2.65i)17-s + (2.23 − 6.87i)18-s + ⋯
L(s)  = 1  + (1.45 − 1.06i)2-s + (−0.0732 − 0.225i)3-s + (0.697 − 2.14i)4-s + (−1.24 − 0.901i)5-s + (−0.346 − 0.251i)6-s + (−0.116 + 0.359i)7-s + (−0.700 − 2.15i)8-s + (0.763 − 0.554i)9-s − 2.76·10-s − 0.534·12-s + (−0.488 + 0.354i)13-s + (0.210 + 0.648i)14-s + (−0.112 + 0.345i)15-s + (−1.48 − 1.07i)16-s + (−0.886 − 0.643i)17-s + (0.526 − 1.61i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.995 - 0.0913i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.995 - 0.0913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113434 + 2.47718i\)
\(L(\frac12)\) \(\approx\) \(0.113434 + 2.47718i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-2.06 + 1.49i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.126 + 0.390i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.77 + 2.01i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (1.75 - 1.27i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.65 + 2.65i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.748 - 2.30i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.648T + 23T^{2} \)
29 \( 1 + (-0.387 + 1.19i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.50 + 4.72i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.54 + 4.76i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.810 - 2.49i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.46T + 43T^{2} \)
47 \( 1 + (1.55 + 4.79i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (10.8 - 7.87i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.37 + 7.29i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-11.5 - 8.42i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.22T + 67T^{2} \)
71 \( 1 + (3.42 + 2.48i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.83 + 5.63i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.89 + 5.73i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.77 + 4.92i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 4.76T + 89T^{2} \)
97 \( 1 + (-7.04 + 5.11i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.913704514928248949149499187260, −9.191294763774479750223633221903, −7.989964141525317968653983545425, −6.94431043859038078240893074730, −5.96664015503683106922831771998, −4.76796177801576894826937957445, −4.35508782181842622766827997352, −3.46071607847051879568556561026, −2.18052717455140290389212423143, −0.76935163373594220574888829748, 2.73717573270884576272189257844, 3.68366042023239330572716531050, 4.42301565107045833383088678581, 5.10484554586925582890811627132, 6.54638496369667192915901967063, 6.92840126395365318330694899439, 7.73173398408175839167096909169, 8.341712649865707732789644256126, 9.976913048692759770795750585853, 10.92524386346114961803065135249

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