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Properties

Label 2-847-77.76-c1-0-58
Degree $2$
Conductor $847$
Sign $-0.916 - 0.400i$
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  − 1.22i·2-s − 1.30i·3-s + 0.488·4-s − 2.38i·5-s − 1.60·6-s + (−2.59 − 0.502i)7-s − 3.05i·8-s + 1.29·9-s − 2.93·10-s − 0.638i·12-s − 2.59·13-s + (−0.618 + 3.19i)14-s − 3.11·15-s − 2.78·16-s − 4.53·17-s − 1.59i·18-s + ⋯
L(s)  = 1  − 0.869i·2-s − 0.753i·3-s + 0.244·4-s − 1.06i·5-s − 0.655·6-s + (−0.981 − 0.190i)7-s − 1.08i·8-s + 0.431·9-s − 0.927·10-s − 0.184i·12-s − 0.720·13-s + (−0.165 + 0.853i)14-s − 0.804·15-s − 0.695·16-s − 1.10·17-s − 0.375i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.916 - 0.400i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (846, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.916 - 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.291757 + 1.39607i\)
\(L(\frac12)\) \(\approx\) \(0.291757 + 1.39607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.59 + 0.502i)T \)
11 \( 1 \)
good2 \( 1 + 1.22iT - 2T^{2} \)
3 \( 1 + 1.30iT - 3T^{2} \)
5 \( 1 + 2.38iT - 5T^{2} \)
13 \( 1 + 2.59T + 13T^{2} \)
17 \( 1 + 4.53T + 17T^{2} \)
19 \( 1 - 8.22T + 19T^{2} \)
23 \( 1 + 3.85T + 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 - 4.13iT - 31T^{2} \)
37 \( 1 - 4.02T + 37T^{2} \)
41 \( 1 + 0.808T + 41T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 + 1.84iT - 47T^{2} \)
53 \( 1 + 2.50T + 53T^{2} \)
59 \( 1 + 10.7iT - 59T^{2} \)
61 \( 1 + 6.21T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 9.88T + 71T^{2} \)
73 \( 1 - 4.78T + 73T^{2} \)
79 \( 1 - 4.82iT - 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 1.52iT - 89T^{2} \)
97 \( 1 - 10.2iT - 97T^{2} \)
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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

−9.688017043784488671529073856169, −9.234715294417098855113896179209, −7.922523821717805082975765651724, −7.06956707376059075321180384035, −6.46406594676125698347567899873, −5.15801627888562971579260661914, −4.04569821520637082796952543310, −2.92211600235918011125684640533, −1.75274120222637926102694752958, −0.67619163245259141926013611152, 2.40498540878708646120059359416, 3.28848762332690350783542836970, 4.51084731807983844171471054254, 5.63945286449329952225878119485, 6.45299190833450476270079776865, 7.15903225034511792019811125604, 7.76477102709727756192346344014, 9.166360844537792632880237156778, 9.791076335547221517249330316953, 10.49998378873806598652153911565

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