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Properties

Label 2-845-65.29-c1-0-26
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: $\chi_{845} (484, \cdot )$
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

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

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