Properties

Label 2-845-65.9-c1-0-65
Degree $2$
Conductor $845$
Sign $-0.561 - 0.827i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (−1.73 − i)3-s + (−0.500 − 0.866i)4-s + (−2 − i)5-s + (−0.999 − 1.73i)6-s − 3i·8-s + (0.499 + 0.866i)9-s + (−1.23 − 1.86i)10-s + (−1 + 1.73i)11-s + 2i·12-s + (2.46 + 3.73i)15-s + (0.500 − 0.866i)16-s + 0.999i·18-s + (−3 − 5.19i)19-s + (0.133 + 2.23i)20-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.999 − 0.577i)3-s + (−0.250 − 0.433i)4-s + (−0.894 − 0.447i)5-s + (−0.408 − 0.707i)6-s − 1.06i·8-s + (0.166 + 0.288i)9-s + (−0.389 − 0.590i)10-s + (−0.301 + 0.522i)11-s + 0.577i·12-s + (0.636 + 0.963i)15-s + (0.125 − 0.216i)16-s + 0.235i·18-s + (−0.688 − 1.19i)19-s + (0.0299 + 0.499i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.561 - 0.827i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.561 - 0.827i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2 + i)T \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.73 + i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + (-5.19 - 3i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4 - 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.19 + 3i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.3 + 6i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1 + 1.73i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + (-4 + 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.19 - 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.543789780690554871655209599776, −8.858719341934926572582871697203, −7.52095306459969370321986249112, −6.93519439897142374017504170268, −6.08429378801324898402518469454, −5.06618025674597018161403539745, −4.65979668096154567213864514349, −3.36254859029424743067453971620, −1.28271464041163100241285931310, 0, 2.56972880441401852347367313286, 3.74190933382648334906712772201, 4.32206367353055903962644824527, 5.32196058236064618725849255337, 6.09032733714847938142265197854, 7.38286258226968259286566103250, 8.158426651409492409744277923309, 9.017820548054649837473707076068, 10.43709436307499433198559531670

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