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Properties

Label 2-656-41.18-c1-0-16
Degree $2$
Conductor $656$
Sign $-0.853 + 0.521i$
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  − 0.536·3-s + (1.18 − 0.864i)5-s + (−1.04 + 3.22i)7-s − 2.71·9-s + (−4.18 − 3.03i)11-s + (−1.60 − 4.94i)13-s + (−0.638 + 0.463i)15-s + (0.00992 + 0.00721i)17-s + (−0.532 + 1.63i)19-s + (0.561 − 1.72i)21-s + (−1.34 − 4.15i)23-s + (−0.876 + 2.69i)25-s + 3.06·27-s + (6.38 − 4.63i)29-s + (−6.47 − 4.70i)31-s + ⋯
L(s)  = 1  − 0.309·3-s + (0.532 − 0.386i)5-s + (−0.395 + 1.21i)7-s − 0.904·9-s + (−1.26 − 0.916i)11-s + (−0.445 − 1.37i)13-s + (−0.164 + 0.119i)15-s + (0.00240 + 0.00174i)17-s + (−0.122 + 0.375i)19-s + (0.122 − 0.377i)21-s + (−0.281 − 0.866i)23-s + (−0.175 + 0.539i)25-s + 0.589·27-s + (1.18 − 0.860i)29-s + (−1.16 − 0.845i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $-0.853 + 0.521i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ -0.853 + 0.521i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0998365 - 0.354944i\)
\(L(\frac12)\) \(\approx\) \(0.0998365 - 0.354944i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.865 + 6.34i)T \)
good3 \( 1 + 0.536T + 3T^{2} \)
5 \( 1 + (-1.18 + 0.864i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.04 - 3.22i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (4.18 + 3.03i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.60 + 4.94i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.00992 - 0.00721i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.532 - 1.63i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.34 + 4.15i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-6.38 + 4.63i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (6.47 + 4.70i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (6.51 - 4.73i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (1.75 + 5.40i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-2.40 - 7.39i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (10.6 - 7.71i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.308 - 0.949i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.41 - 7.44i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-12.4 + 9.05i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (10.9 + 7.93i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + 2.99T + 73T^{2} \)
79 \( 1 - 0.897T + 79T^{2} \)
83 \( 1 - 6.26T + 83T^{2} \)
89 \( 1 + (2.00 - 6.17i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.71 - 2.69i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.30378683153220870474873283354, −9.231568343709803799139513412203, −8.467846851052323131095851421533, −7.81038415351550680329945861732, −6.13604064635729697557538260081, −5.67201724499648479116088364483, −5.08045370448310388515840950358, −3.14654475306550751593599668701, −2.42042216180517074804255902083, −0.18639461152410136275564036044, 1.98043906591473476634180157163, 3.21841127936610203591633182090, 4.55477324567864499046905162269, 5.40330577859373082339719982405, 6.70134208391403200464732779134, 7.08184882198461045526912386428, 8.226069752751336732390532862443, 9.397326506309354861704687244769, 10.15055595722020482952139551195, 10.73099669148995359559557958061

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