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Properties

Label 2-847-847.342-c0-0-0
Degree $2$
Conductor $847$
Sign $0.914 + 0.403i$
Analytic cond. $0.422708$
Root an. cond. $0.650160$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.118 − 0.258i)2-s + (0.601 − 0.694i)4-s + (−0.142 + 0.989i)7-s + (−0.524 − 0.153i)8-s + 9-s + (0.415 + 0.909i)11-s + (0.273 − 0.0801i)14-s + (−0.108 − 0.755i)16-s + (−0.118 − 0.258i)18-s + (0.186 − 0.215i)22-s + (0.0405 + 0.281i)23-s + (0.841 − 0.540i)25-s + (0.601 + 0.694i)28-s + (−1.61 − 1.03i)29-s + (−0.642 + 0.412i)32-s + ⋯
L(s)  = 1  + (−0.118 − 0.258i)2-s + (0.601 − 0.694i)4-s + (−0.142 + 0.989i)7-s + (−0.524 − 0.153i)8-s + 9-s + (0.415 + 0.909i)11-s + (0.273 − 0.0801i)14-s + (−0.108 − 0.755i)16-s + (−0.118 − 0.258i)18-s + (0.186 − 0.215i)22-s + (0.0405 + 0.281i)23-s + (0.841 − 0.540i)25-s + (0.601 + 0.694i)28-s + (−1.61 − 1.03i)29-s + (−0.642 + 0.412i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.914 + 0.403i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (342, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.914 + 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.102802354\)
\(L(\frac12)\) \(\approx\) \(1.102802354\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.142 - 0.989i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
good2 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 + (-0.841 + 0.540i)T^{2} \)
13 \( 1 + (0.142 + 0.989i)T^{2} \)
17 \( 1 + (-0.415 + 0.909i)T^{2} \)
19 \( 1 + (-0.415 - 0.909i)T^{2} \)
23 \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \)
29 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
31 \( 1 + (0.142 - 0.989i)T^{2} \)
37 \( 1 + (1.30 + 1.51i)T + (-0.142 + 0.989i)T^{2} \)
41 \( 1 + (0.654 - 0.755i)T^{2} \)
43 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
47 \( 1 + (0.654 + 0.755i)T^{2} \)
53 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
59 \( 1 + (0.654 + 0.755i)T^{2} \)
61 \( 1 + (0.654 + 0.755i)T^{2} \)
67 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (-1.25 + 0.368i)T + (0.841 - 0.540i)T^{2} \)
83 \( 1 + (0.959 + 0.281i)T^{2} \)
89 \( 1 + (-0.415 + 0.909i)T^{2} \)
97 \( 1 + (-0.841 - 0.540i)T^{2} \)
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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

−10.31663860502920900444083737599, −9.493327595063101640374242924288, −9.048467733529567385472146063467, −7.60571113617717097074352934317, −6.89170842414597703942749808937, −6.00517290763287316288627515604, −5.13010429337777360955232345096, −3.95929769460795635013533977062, −2.49329779713884158149694995665, −1.62695862705032066453783647584, 1.54912102598778970747279317216, 3.25452678121079886770695943691, 3.86821169394955749031194947281, 5.14954753704002422058040170502, 6.54472263396915841542243643558, 6.95396051455424349214758405206, 7.77912873028118941438788515847, 8.647267801172744797794944748069, 9.561454510152831493525101492796, 10.62654743106361817010262912002

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