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Properties

Label 2-847-847.174-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  + (1.38 − 0.158i)2-s + (0.918 − 0.213i)4-s + (0.696 − 0.717i)7-s + (−0.0749 + 0.0267i)8-s + (0.309 − 0.951i)9-s + (−0.736 + 0.676i)11-s + (0.851 − 1.10i)14-s + (−0.945 + 0.464i)16-s + (0.276 − 1.36i)18-s + (−0.912 + 1.05i)22-s + (0.132 + 0.923i)23-s + (−0.362 + 0.931i)25-s + (0.486 − 0.807i)28-s + (0.409 + 1.05i)29-s + (−1.16 + 0.751i)32-s + ⋯
L(s)  = 1  + (1.38 − 0.158i)2-s + (0.918 − 0.213i)4-s + (0.696 − 0.717i)7-s + (−0.0749 + 0.0267i)8-s + (0.309 − 0.951i)9-s + (−0.736 + 0.676i)11-s + (0.851 − 1.10i)14-s + (−0.945 + 0.464i)16-s + (0.276 − 1.36i)18-s + (−0.912 + 1.05i)22-s + (0.132 + 0.923i)23-s + (−0.362 + 0.931i)25-s + (0.486 − 0.807i)28-s + (0.409 + 1.05i)29-s + (−1.16 + 0.751i)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: $\chi_{847} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.914 + 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.923213290\)
\(L(\frac12)\) \(\approx\) \(1.923213290\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.696 + 0.717i)T \)
11 \( 1 + (0.736 - 0.676i)T \)
good2 \( 1 + (-1.38 + 0.158i)T + (0.974 - 0.226i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.362 - 0.931i)T^{2} \)
13 \( 1 + (0.466 - 0.884i)T^{2} \)
17 \( 1 + (-0.198 - 0.980i)T^{2} \)
19 \( 1 + (-0.993 + 0.113i)T^{2} \)
23 \( 1 + (-0.132 - 0.923i)T + (-0.959 + 0.281i)T^{2} \)
29 \( 1 + (-0.409 - 1.05i)T + (-0.736 + 0.676i)T^{2} \)
31 \( 1 + (-0.696 + 0.717i)T^{2} \)
37 \( 1 + (0.138 + 1.61i)T + (-0.985 + 0.170i)T^{2} \)
41 \( 1 + (-0.516 - 0.856i)T^{2} \)
43 \( 1 + (1.72 + 0.505i)T + (0.841 + 0.540i)T^{2} \)
47 \( 1 + (0.921 - 0.389i)T^{2} \)
53 \( 1 + (0.456 + 0.224i)T + (0.610 + 0.791i)T^{2} \)
59 \( 1 + (-0.516 + 0.856i)T^{2} \)
61 \( 1 + (-0.974 - 0.226i)T^{2} \)
67 \( 1 + (-0.643 - 1.40i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.262 - 0.998i)T + (-0.870 - 0.491i)T^{2} \)
73 \( 1 + (0.564 + 0.825i)T^{2} \)
79 \( 1 + (-0.0526 + 1.84i)T + (-0.998 - 0.0570i)T^{2} \)
83 \( 1 + (-0.941 + 0.336i)T^{2} \)
89 \( 1 + (-0.415 + 0.909i)T^{2} \)
97 \( 1 + (0.362 + 0.931i)T^{2} \)
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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

−10.56167449833897864237762199956, −9.636552338973897776039644491515, −8.655596700662655879896016148683, −7.43198159397663346328520924735, −6.84916970814118275442406267258, −5.61489285809855074750449709065, −4.94052634023123652006749300962, −4.00733752069846742234467090199, −3.25820842937008989527728015250, −1.77437286204729301918403284724, 2.19305383366908369993138888752, 3.08005031451364925227173215088, 4.51519249081722225314371704184, 4.95875988931853271825627774772, 5.86164524975289445768666615808, 6.65214838669322478158253089868, 8.018933839570337539945819060553, 8.394432778775495147866881396127, 9.746320314646220856333984607637, 10.69646753435004745951946532091

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