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Properties

Label 2-847-11.4-c1-0-4
Degree $2$
Conductor $847$
Sign $-0.530 - 0.847i$
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  + (−1.18 − 0.862i)2-s + (−0.5 + 1.53i)3-s + (0.0467 + 0.143i)4-s + (−0.377 + 0.274i)5-s + (1.91 − 1.39i)6-s + (0.309 + 0.951i)7-s + (−0.837 + 2.57i)8-s + (0.309 + 0.224i)9-s + 0.684·10-s − 0.244·12-s + (−1.28 − 0.930i)13-s + (0.453 − 1.39i)14-s + (−0.233 − 0.718i)15-s + (3.46 − 2.51i)16-s + (4.22 − 3.07i)17-s + (−0.173 − 0.532i)18-s + ⋯
L(s)  = 1  + (−0.839 − 0.609i)2-s + (−0.288 + 0.888i)3-s + (0.0233 + 0.0719i)4-s + (−0.168 + 0.122i)5-s + (0.783 − 0.569i)6-s + (0.116 + 0.359i)7-s + (−0.296 + 0.911i)8-s + (0.103 + 0.0748i)9-s + 0.216·10-s − 0.0706·12-s + (−0.355 − 0.257i)13-s + (0.121 − 0.372i)14-s + (−0.0602 − 0.185i)15-s + (0.865 − 0.628i)16-s + (1.02 − 0.745i)17-s + (−0.0408 − 0.125i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.530 - 0.847i$
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: \((2,\ 847,\ (\ :1/2),\ -0.530 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.218426 + 0.394181i\)
\(L(\frac12)\) \(\approx\) \(0.218426 + 0.394181i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 + 0.862i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.5 - 1.53i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (0.377 - 0.274i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (1.28 + 0.930i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.22 + 3.07i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.30 - 4.01i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.80T + 23T^{2} \)
29 \( 1 + (0.840 + 2.58i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.04 + 0.760i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.600 + 1.84i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.321 + 0.990i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.70T + 43T^{2} \)
47 \( 1 + (1.97 - 6.08i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-10.6 - 7.76i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.65 + 8.18i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (12.3 - 8.95i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 4.67T + 67T^{2} \)
71 \( 1 + (7.88 - 5.72i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.11 - 12.6i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.89 - 2.10i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (13.9 - 10.1i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 8.91T + 89T^{2} \)
97 \( 1 + (-2.18 - 1.58i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.25635440106630469776947523884, −9.832136788719011564174148862117, −9.151133694016272311793741095224, −8.123689150442779750063459037188, −7.38677781519963495012904873768, −5.77926377934428951658634327515, −5.25869400269511518903770958094, −4.10131385517307255185948737741, −2.88795431524597373984807108043, −1.54648428541411035824280113533, 0.31580552227166787671855171592, 1.65273974384312752615958284109, 3.42200320398462007584817329938, 4.53593908881362399947244421544, 5.95789842275487614982917029154, 6.74653037732416421094550924787, 7.37076092471693368006457106379, 8.069949360536006305847227196509, 8.811857455336198427076384471704, 9.827796553886033000897516777188

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