Properties

Label 2-845-65.9-c1-0-22
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 $0$

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: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.561 + 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.743055 - 0.393679i\)
\(L(\frac12)\) \(\approx\) \(0.743055 - 0.393679i\)
\(L(\frac{3}{2})\) 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

−10.14999570423228012719315290655, −9.363166610746784435528916459209, −8.637703992207426307066825157727, −7.35461599111892442182795793747, −6.49826836846500585277619445567, −5.61483506756877843898924916074, −5.26027619252532551704857829170, −3.39368124992045357126077925450, −1.88131942843778828910638235597, −0.908918670025635236534274206943, 0.867183432233262071665442271527, 2.77216192227404656028162942212, 4.43509935186429329320679287075, 4.90096552373643768927047908376, 6.04858793039101799910238566163, 6.78983390011302134856340565321, 7.83123570605859280238897725072, 8.865490452114674271522852528368, 9.503277976966120547356347831206, 10.07280789414748847679526620163

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