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Properties

Label 2-847-847.454-c0-0-0
Degree $2$
Conductor $847$
Sign $0.995 + 0.0933i$
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  + (1.83 − 0.105i)2-s + (2.37 − 0.272i)4-s + (−0.921 + 0.389i)7-s + (2.53 − 0.437i)8-s + (−0.809 + 0.587i)9-s + (−0.362 − 0.931i)11-s + (−1.65 + 0.812i)14-s + (2.27 − 0.529i)16-s + (−1.42 + 1.16i)18-s + (−0.765 − 1.67i)22-s + (−0.676 + 0.780i)23-s + (−0.564 − 0.825i)25-s + (−2.08 + 1.17i)28-s + (0.526 − 0.770i)29-s + (1.66 − 0.489i)32-s + ⋯
L(s)  = 1  + (1.83 − 0.105i)2-s + (2.37 − 0.272i)4-s + (−0.921 + 0.389i)7-s + (2.53 − 0.437i)8-s + (−0.809 + 0.587i)9-s + (−0.362 − 0.931i)11-s + (−1.65 + 0.812i)14-s + (2.27 − 0.529i)16-s + (−1.42 + 1.16i)18-s + (−0.765 − 1.67i)22-s + (−0.676 + 0.780i)23-s + (−0.564 − 0.825i)25-s + (−2.08 + 1.17i)28-s + (0.526 − 0.770i)29-s + (1.66 − 0.489i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.995 + 0.0933i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (454, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.995 + 0.0933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.405604021\)
\(L(\frac12)\) \(\approx\) \(2.405604021\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.921 - 0.389i)T \)
11 \( 1 + (0.362 + 0.931i)T \)
good2 \( 1 + (-1.83 + 0.105i)T + (0.993 - 0.113i)T^{2} \)
3 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (0.564 + 0.825i)T^{2} \)
13 \( 1 + (-0.516 - 0.856i)T^{2} \)
17 \( 1 + (-0.774 - 0.633i)T^{2} \)
19 \( 1 + (0.998 - 0.0570i)T^{2} \)
23 \( 1 + (0.676 - 0.780i)T + (-0.142 - 0.989i)T^{2} \)
29 \( 1 + (-0.526 + 0.770i)T + (-0.362 - 0.931i)T^{2} \)
31 \( 1 + (0.921 - 0.389i)T^{2} \)
37 \( 1 + (0.455 + 0.417i)T + (0.0855 + 0.996i)T^{2} \)
41 \( 1 + (0.870 + 0.491i)T^{2} \)
43 \( 1 + (0.277 - 1.92i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (-0.198 - 0.980i)T^{2} \)
53 \( 1 + (-1.18 - 0.276i)T + (0.897 + 0.441i)T^{2} \)
59 \( 1 + (0.870 - 0.491i)T^{2} \)
61 \( 1 + (-0.993 - 0.113i)T^{2} \)
67 \( 1 + (-1.58 - 1.01i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (1.06 + 1.37i)T + (-0.254 + 0.967i)T^{2} \)
73 \( 1 + (0.466 - 0.884i)T^{2} \)
79 \( 1 + (-0.276 - 0.284i)T + (-0.0285 + 0.999i)T^{2} \)
83 \( 1 + (0.985 - 0.170i)T^{2} \)
89 \( 1 + (-0.841 + 0.540i)T^{2} \)
97 \( 1 + (0.564 - 0.825i)T^{2} \)
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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

−10.71804302189328180748107441621, −9.836277832650182110097209061661, −8.517036602197532234755164764610, −7.60185689382165262159837633966, −6.34486794048853686603122592944, −5.91248882248428957401906328457, −5.15299141570181245495127877538, −3.97923141214295948062375910190, −3.06693460383367104116903354859, −2.33024921093041238295212383426, 2.25626164444650681522156015749, 3.29197535829165376273821712819, 4.00498651669081962388987088775, 5.09417726269625614551158464891, 5.89043392343405331908677881228, 6.74149514209798496012888862977, 7.29842666007102668614085930194, 8.611234035499099646771240162385, 9.854924560801412253075951022452, 10.60293103899666235179243871712

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