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Properties

Label 2-847-11.4-c1-0-51
Degree $2$
Conductor $847$
Sign $-0.898 + 0.437i$
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  + (0.0973 + 0.0707i)2-s + (0.855 − 2.63i)3-s + (−0.613 − 1.88i)4-s + (2.27 − 1.65i)5-s + (0.269 − 0.195i)6-s + (−0.309 − 0.951i)7-s + (0.148 − 0.456i)8-s + (−3.77 − 2.73i)9-s + 0.338·10-s − 5.49·12-s + (0.873 + 0.634i)13-s + (0.0371 − 0.114i)14-s + (−2.40 − 7.39i)15-s + (−3.16 + 2.30i)16-s + (5.62 − 4.08i)17-s + (−0.173 − 0.533i)18-s + ⋯
L(s)  = 1  + (0.0688 + 0.0500i)2-s + (0.493 − 1.51i)3-s + (−0.306 − 0.944i)4-s + (1.01 − 0.738i)5-s + (0.110 − 0.0799i)6-s + (−0.116 − 0.359i)7-s + (0.0524 − 0.161i)8-s + (−1.25 − 0.913i)9-s + 0.106·10-s − 1.58·12-s + (0.242 + 0.176i)13-s + (0.00994 − 0.0305i)14-s + (−0.620 − 1.90i)15-s + (−0.791 + 0.575i)16-s + (1.36 − 0.991i)17-s + (−0.0408 − 0.125i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.898 + 0.437i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.898 + 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.474128 - 2.05569i\)
\(L(\frac12)\) \(\approx\) \(0.474128 - 2.05569i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0973 - 0.0707i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.855 + 2.63i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-2.27 + 1.65i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-0.873 - 0.634i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-5.62 + 4.08i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.33 - 7.17i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + (0.379 + 1.16i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-6.53 - 4.74i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.474 - 1.46i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.87 - 8.84i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.23T + 43T^{2} \)
47 \( 1 + (0.585 - 1.80i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.09 - 2.24i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.05 + 6.33i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.92 - 5.76i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.06T + 67T^{2} \)
71 \( 1 + (10.1 - 7.35i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.793 + 2.44i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (12.4 + 9.00i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.65 + 1.20i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.76T + 89T^{2} \)
97 \( 1 + (7.37 + 5.35i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.761564299463893452415714621265, −8.954885264088482196709235345125, −8.141910775293307363032345658333, −7.20448707400616803729934424193, −6.24352615707955737127366478425, −5.71133941422763726483205295093, −4.65648894301510786241519174415, −2.96161960127882160286614375432, −1.55116771003110461519879258919, −1.10535473792811597717765283109, 2.52673591876202278262263163119, 3.14497294463396201783260873912, 4.10027070353561576159978900467, 5.06794178996560831142343774204, 6.02787668766465085449448191801, 7.19579605752912653614668307623, 8.369822328472014541225626231298, 9.005031532680952656289491652392, 9.655161607655174674801254435608, 10.45622605625224883921838816021

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