Properties

Label 2-847-77.58-c1-0-26
Degree $2$
Conductor $847$
Sign $-0.985 + 0.169i$
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  + (0.260 + 2.47i)2-s + (−0.477 + 0.530i)3-s + (−4.11 + 0.874i)4-s + (2.01 − 0.896i)5-s + (−1.43 − 1.04i)6-s + (1.94 + 1.79i)7-s + (−1.69 − 5.22i)8-s + (0.260 + 2.47i)9-s + (2.74 + 4.75i)10-s + (1.5 − 2.59i)12-s + (2.65 − 1.93i)13-s + (−3.93 + 5.28i)14-s + (−0.486 + 1.49i)15-s + (4.81 − 2.14i)16-s + (−0.155 + 1.48i)17-s + (−6.06 + 1.28i)18-s + ⋯
L(s)  = 1  + (0.184 + 1.75i)2-s + (−0.275 + 0.306i)3-s + (−2.05 + 0.437i)4-s + (0.900 − 0.400i)5-s + (−0.587 − 0.426i)6-s + (0.735 + 0.677i)7-s + (−0.599 − 1.84i)8-s + (0.0867 + 0.825i)9-s + (0.868 + 1.50i)10-s + (0.433 − 0.749i)12-s + (0.737 − 0.535i)13-s + (−1.05 + 1.41i)14-s + (−0.125 + 0.386i)15-s + (1.20 − 0.535i)16-s + (−0.0377 + 0.359i)17-s + (−1.43 + 0.304i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.985 + 0.169i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (366, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148454 - 1.73738i\)
\(L(\frac12)\) \(\approx\) \(0.148454 - 1.73738i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.94 - 1.79i)T \)
11 \( 1 \)
good2 \( 1 + (-0.260 - 2.47i)T + (-1.95 + 0.415i)T^{2} \)
3 \( 1 + (0.477 - 0.530i)T + (-0.313 - 2.98i)T^{2} \)
5 \( 1 + (-2.01 + 0.896i)T + (3.34 - 3.71i)T^{2} \)
13 \( 1 + (-2.65 + 1.93i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.155 - 1.48i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (-6.76 - 1.43i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (3.24 - 5.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.509 - 1.56i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.14 + 0.955i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (3.71 + 4.12i)T + (-3.86 + 36.7i)T^{2} \)
41 \( 1 + (3.47 + 10.6i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.26T + 43T^{2} \)
47 \( 1 + (1.45 + 0.309i)T + (42.9 + 19.1i)T^{2} \)
53 \( 1 + (-0.278 - 0.123i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (-12.3 + 2.63i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (11.8 - 5.28i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (-2.28 - 3.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.16 + 6.65i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-8.38 + 1.78i)T + (66.6 - 29.6i)T^{2} \)
79 \( 1 + (0.484 + 4.60i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-1.56 - 1.13i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.60 - 2.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.50 - 1.09i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.39405246144735690769435946884, −9.426883322784079665332429588781, −8.777840137419966008096563556243, −7.902572146590895696281773547512, −7.32906103672163367010675462705, −5.82978652560675452759387056214, −5.60284477557450330978156293435, −5.03920634394180725254980084490, −3.78046150185698927918345675625, −1.78167800268683264724907180880, 0.926026946143007062497285684337, 1.82385814779925985957632399686, 3.02898778764346182732951112745, 4.02218534772722261821929763536, 4.98199624850354064032195317940, 6.11495980806135867141880649660, 7.05754544770113602678142604672, 8.383689483920605128965405462039, 9.402975078462248608100259323521, 9.929012371582833902211503333036

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