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Properties

Label 2-847-11.9-c1-0-49
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

\(L(1)\) \(\approx\) \(0.468101 - 3.18144i\)
\(L(\frac12)\) \(\approx\) \(0.468101 - 3.18144i\)
\(L(\frac{3}{2})\) 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

−9.740705340429444407291600189314, −9.039360234734024848480234182623, −8.404798047433620619308809314018, −7.11639177296989632830977757543, −6.74795969215389831599714279190, −4.87933385639405101186956902235, −3.94233016197365304481896798159, −2.82664784027621376876732786567, −2.27980395868283393910415239732, −1.32562194773685813556297647417, 2.32472013418674254502329423353, 3.56466002821050247600660485455, 4.49029477837385073906260067679, 5.06911163369711263301515057317, 6.20212763506367779221176803904, 7.39542496088911556844974681929, 8.106112242865521234132028135778, 8.553044622632738147644279920599, 9.394736719541027870030756425269, 10.44491428857724212993942769128

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