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Properties

Label 2-847-11.5-c1-0-45
Degree $2$
Conductor $847$
Sign $-0.998 + 0.0475i$
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  + (−0.711 − 2.19i)2-s + (1.86 + 1.35i)3-s + (−2.67 + 1.94i)4-s + (1.11 − 3.42i)5-s + (1.63 − 5.04i)6-s + (−0.809 + 0.587i)7-s + (2.42 + 1.76i)8-s + (0.711 + 2.19i)9-s − 8.30·10-s − 7.60·12-s + (−2.04 − 6.28i)13-s + (1.86 + 1.35i)14-s + (6.71 − 4.88i)15-s + (0.0935 − 0.287i)16-s + (−0.833 + 2.56i)17-s + (4.29 − 3.11i)18-s + ⋯
L(s)  = 1  + (−0.503 − 1.54i)2-s + (1.07 + 0.781i)3-s + (−1.33 + 0.970i)4-s + (0.498 − 1.53i)5-s + (0.668 − 2.05i)6-s + (−0.305 + 0.222i)7-s + (0.858 + 0.623i)8-s + (0.237 + 0.730i)9-s − 2.62·10-s − 2.19·12-s + (−0.566 − 1.74i)13-s + (0.497 + 0.361i)14-s + (1.73 − 1.26i)15-s + (0.0233 − 0.0719i)16-s + (−0.202 + 0.622i)17-s + (1.01 − 0.734i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.998 + 0.0475i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.998 + 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0326933 - 1.37491i\)
\(L(\frac12)\) \(\approx\) \(0.0326933 - 1.37491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.711 + 2.19i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.86 - 1.35i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-1.11 + 3.42i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (2.04 + 6.28i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.833 - 2.56i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.42 + 1.76i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2.69T + 23T^{2} \)
29 \( 1 + (-3.80 + 2.76i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.21 + 3.06i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.66 - 4.11i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.69T + 43T^{2} \)
47 \( 1 + (-1.54 - 1.12i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.98 + 12.2i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.41 - 3.93i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.32 + 4.09i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 8.51T + 67T^{2} \)
71 \( 1 + (1.32 - 4.09i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.04 - 2.93i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.56 - 7.89i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.927 - 2.85i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + (1.11 + 3.42i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.862905800119413351401705827032, −9.185303448577285646763458507187, −8.420942077586696649112512637603, −8.099358192988246750418809810556, −5.98516401752066650674794733328, −4.82579976047723817443995769037, −4.02556688116527420809139997008, −2.97593344118493940666863206729, −2.15004471328163493229844300439, −0.67400751970587116310113809016, 2.04230052972315135588477729020, 2.96433009750334683953563148762, 4.45510331621139530009852795959, 6.03269219953210015869048989515, 6.64332163762551244370965141624, 7.18535471618692427207728662905, 7.71773393421553636699165456999, 8.770404011606219850339958847515, 9.413808398967922674955620378891, 10.15523295144674555689342553510

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