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Properties

Label 2-847-11.9-c1-0-38
Degree $2$
Conductor $847$
Sign $0.957 - 0.288i$
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  + (−0.656 + 2.02i)2-s + (2.52 − 1.83i)3-s + (−2.03 − 1.47i)4-s + (0.149 + 0.461i)5-s + (2.05 + 6.31i)6-s + (−0.809 − 0.587i)7-s + (0.885 − 0.643i)8-s + (2.09 − 6.43i)9-s − 1.03·10-s − 7.85·12-s + (1.73 − 5.33i)13-s + (1.71 − 1.24i)14-s + (1.22 + 0.890i)15-s + (−0.835 − 2.57i)16-s + (1.73 + 5.33i)17-s + (11.6 + 8.44i)18-s + ⋯
L(s)  = 1  + (−0.464 + 1.42i)2-s + (1.45 − 1.06i)3-s + (−1.01 − 0.739i)4-s + (0.0670 + 0.206i)5-s + (0.837 + 2.57i)6-s + (−0.305 − 0.222i)7-s + (0.313 − 0.227i)8-s + (0.696 − 2.14i)9-s − 0.325·10-s − 2.26·12-s + (0.480 − 1.47i)13-s + (0.459 − 0.333i)14-s + (0.316 + 0.229i)15-s + (−0.208 − 0.642i)16-s + (0.420 + 1.29i)17-s + (2.74 + 1.99i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.957 - 0.288i$
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: \((2,\ 847,\ (\ :1/2),\ 0.957 - 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91081 + 0.281148i\)
\(L(\frac12)\) \(\approx\) \(1.91081 + 0.281148i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.656 - 2.02i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-2.52 + 1.83i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.149 - 0.461i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-1.73 + 5.33i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.73 - 5.33i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.27 + 3.10i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 2.48T + 23T^{2} \)
29 \( 1 + (4.27 + 3.10i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.20 - 6.77i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.190 - 0.138i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.93 - 1.40i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.03T + 43T^{2} \)
47 \( 1 + (-1.30 + 0.946i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.936 - 2.88i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.52 + 1.83i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.738 + 2.27i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + (-3.73 - 11.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.93 + 1.40i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.79 + 8.58i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.994 + 3.06i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 1.26T + 89T^{2} \)
97 \( 1 + (2.71 - 8.36i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.740339249655124635349612172665, −8.932528781647909442120614198934, −8.262500392742676359021989223628, −7.74367755895701711644336607132, −7.01540221233523609180557318289, −6.33528873030562299483110182603, −5.35024499702764433508283120387, −3.55623129207047074436850138974, −2.75063736078083602508773108103, −1.04353675344503799419971434686, 1.59187539822257704758407332164, 2.67338276232110242397823983828, 3.44740429524669653599048439183, 4.16709048610223201793035892315, 5.30234249542531605812400912328, 7.03643812839233974375699493724, 8.125704629203993380557118887467, 9.163511853016534585583420135735, 9.285864093430629265373203306927, 9.846685423282313811388187730038

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