Properties

Label 2-847-11.4-c1-0-50
Degree $2$
Conductor $847$
Sign $-0.975 + 0.220i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.975 + 0.220i$
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: \((2,\ 847,\ (\ :1/2),\ -0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.200030 - 1.79052i\)
\(L(\frac12)\) \(\approx\) \(0.200030 - 1.79052i\)
\(L(\frac{3}{2})\) 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

−9.495525912357928564578385860394, −8.868182464696661889977127938669, −8.403455184065854105646198368660, −6.97244518592565492235892502344, −6.46888518011872726514431014397, −5.46910164406622925355976763154, −4.73456296434903427415608238054, −2.64099378909664532754990613949, −1.82350934120853881587965453824, −0.871429173068083778365286554980, 2.45867546859529019739475159983, 3.23123154621193278879915429828, 4.24223478737585987155137583340, 4.94788534785741594170878483377, 6.23673362561087439476601050410, 7.26872239266081058520833260153, 8.255781450358307615482277013622, 9.136458952611594361396319294562, 9.723158189213413209717548467349, 10.27945445161082950965706633547

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