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Properties

Label 2-656-41.16-c1-0-3
Degree $2$
Conductor $656$
Sign $0.898 - 0.439i$
Analytic cond. $5.23818$
Root an. cond. $2.28870$
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  − 1.92·3-s + (−3.33 − 2.42i)5-s + (0.669 + 2.06i)7-s + 0.721·9-s + (−2.97 + 2.16i)11-s + (0.806 − 2.48i)13-s + (6.42 + 4.66i)15-s + (1.33 − 0.966i)17-s + (−0.968 − 2.98i)19-s + (−1.29 − 3.97i)21-s + (−1.16 + 3.58i)23-s + (3.69 + 11.3i)25-s + 4.39·27-s + (6.94 + 5.04i)29-s + (7.05 − 5.12i)31-s + ⋯
L(s)  = 1  − 1.11·3-s + (−1.48 − 1.08i)5-s + (0.253 + 0.778i)7-s + 0.240·9-s + (−0.897 + 0.652i)11-s + (0.223 − 0.688i)13-s + (1.65 + 1.20i)15-s + (0.322 − 0.234i)17-s + (−0.222 − 0.683i)19-s + (−0.281 − 0.867i)21-s + (−0.242 + 0.746i)23-s + (0.738 + 2.27i)25-s + 0.846·27-s + (1.29 + 0.937i)29-s + (1.26 − 0.920i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $0.898 - 0.439i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ 0.898 - 0.439i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.537737 + 0.124451i\)
\(L(\frac12)\) \(\approx\) \(0.537737 + 0.124451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 + (4.88 + 4.14i)T \)
good3 \( 1 + 1.92T + 3T^{2} \)
5 \( 1 + (3.33 + 2.42i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.669 - 2.06i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (2.97 - 2.16i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.806 + 2.48i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.33 + 0.966i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.968 + 2.98i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.16 - 3.58i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-6.94 - 5.04i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-7.05 + 5.12i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-6.25 - 4.54i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-0.357 + 1.09i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (2.21 - 6.82i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.36 + 3.17i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.04 - 9.36i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.09 - 9.53i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (0.772 + 0.561i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-1.89 + 1.37i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 - 12.6T + 73T^{2} \)
79 \( 1 + 7.14T + 79T^{2} \)
83 \( 1 + 1.54T + 83T^{2} \)
89 \( 1 + (-1.79 - 5.52i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (7.59 + 5.52i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.89095508180509816147240891586, −9.830242971887975404277025039552, −8.584531112365630756311391372655, −8.116026507671973592897581566135, −7.15869724327526576453034762571, −5.83422015083635320358381998835, −5.02531897420734293847189892971, −4.50469047927267691772775847079, −2.93378424238913993578845501487, −0.835177104322801241976803534020, 0.52132598782960532775009067010, 2.89817025180745830294196021831, 3.99879398365442209471330135059, 4.84256835053014454141326538761, 6.26602475230598497233094867148, 6.72725503543427752441939189024, 7.949047647462904717220816943357, 8.251446568588478902707834911468, 10.18650551774013775140591426250, 10.63838623724059150781570600289

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