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Properties

Label 2-847-11.4-c1-0-46
Degree $2$
Conductor $847$
Sign $-0.266 - 0.963i$
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  + (−1.80 − 1.31i)2-s + (1 − 3.07i)3-s + (0.927 + 2.85i)4-s + (1.61 − 1.17i)5-s + (−5.85 + 4.25i)6-s + (−0.309 − 0.951i)7-s + (0.690 − 2.12i)8-s + (−6.04 − 4.39i)9-s − 4.47·10-s + 9.70·12-s + (−1 − 0.726i)13-s + (−0.690 + 2.12i)14-s + (−2 − 6.15i)15-s + (0.809 − 0.587i)16-s + (1 − 0.726i)17-s + (5.16 + 15.8i)18-s + ⋯
L(s)  = 1  + (−1.27 − 0.929i)2-s + (0.577 − 1.77i)3-s + (0.463 + 1.42i)4-s + (0.723 − 0.525i)5-s + (−2.38 + 1.73i)6-s + (−0.116 − 0.359i)7-s + (0.244 − 0.751i)8-s + (−2.01 − 1.46i)9-s − 1.41·10-s + 2.80·12-s + (−0.277 − 0.201i)13-s + (−0.184 + 0.568i)14-s + (−0.516 − 1.58i)15-s + (0.202 − 0.146i)16-s + (0.242 − 0.176i)17-s + (1.21 + 3.74i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.266 - 0.963i$
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: \((2,\ 847,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.485111 + 0.637729i\)
\(L(\frac12)\) \(\approx\) \(0.485111 + 0.637729i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 + 1.31i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-1 + 3.07i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.61 + 1.17i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (1 + 0.726i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1 + 0.726i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.763 + 2.35i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 + (-0.145 - 0.449i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.85 - 4.25i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.145 - 0.449i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.09 + 6.43i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-2.23 + 6.88i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (6.85 + 4.97i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1 - 3.07i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.23 - 1.62i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 5.52T + 67T^{2} \)
71 \( 1 + (-1.23 + 0.898i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.61 - 4.97i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.23 - 5.25i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-12.4 + 9.06i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + (-7.61 - 5.53i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.514415009311939669021536001268, −8.704408052343327731738367895348, −8.117867486651597255175548256309, −7.36395141035741389167215403531, −6.50616355943612034167987669765, −5.38332104307928491092898709794, −3.34663882314299484026709262457, −2.32065376511929835661828790468, −1.57371022906192222282030887624, −0.54420762736703976741782352982, 2.24692234904594493047724476117, 3.46560500483535225992482693794, 4.65784773074121400993553468477, 5.83488804446860787736878616851, 6.35213895544602688783696046210, 7.891964678286185120076520369753, 8.279151388154642274254787080884, 9.415264311942528232791721890432, 9.661904217952648679591147180796, 10.23653214742130596922718819202

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