Properties

Label 2-406-7.2-c1-0-16
Degree $2$
Conductor $406$
Sign $-0.821 + 0.569i$
Analytic cond. $3.24192$
Root an. cond. $1.80053$
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  + (−0.5 + 0.866i)2-s + (−1.20 − 2.08i)3-s + (−0.499 − 0.866i)4-s + (1.10 − 1.91i)5-s + 2.40·6-s + (−1.36 − 2.26i)7-s + 0.999·8-s + (−1.39 + 2.41i)9-s + (1.10 + 1.91i)10-s + (0.593 + 1.02i)11-s + (−1.20 + 2.08i)12-s + 1.24·13-s + (2.64 − 0.0506i)14-s − 5.31·15-s + (−0.5 + 0.866i)16-s + (−2.98 − 5.16i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.694 − 1.20i)3-s + (−0.249 − 0.433i)4-s + (0.493 − 0.855i)5-s + 0.982·6-s + (−0.516 − 0.856i)7-s + 0.353·8-s + (−0.465 + 0.806i)9-s + (0.349 + 0.604i)10-s + (0.179 + 0.310i)11-s + (−0.347 + 0.601i)12-s + 0.344·13-s + (0.706 − 0.0135i)14-s − 1.37·15-s + (−0.125 + 0.216i)16-s + (−0.723 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(406\)    =    \(2 \cdot 7 \cdot 29\)
Sign: $-0.821 + 0.569i$
Analytic conductor: \(3.24192\)
Root analytic conductor: \(1.80053\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{406} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 406,\ (\ :1/2),\ -0.821 + 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.195623 - 0.625629i\)
\(L(\frac12)\) \(\approx\) \(0.195623 - 0.625629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (1.36 + 2.26i)T \)
29 \( 1 + T \)
good3 \( 1 + (1.20 + 2.08i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.10 + 1.91i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.593 - 1.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.24T + 13T^{2} \)
17 \( 1 + (2.98 + 5.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.09 - 1.90i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.87 - 6.70i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (4.65 + 8.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.45 + 5.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 5.76T + 43T^{2} \)
47 \( 1 + (-2.91 + 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.840 + 1.45i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.98 - 6.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.05 - 7.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.07 + 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.61T + 71T^{2} \)
73 \( 1 + (0.0197 + 0.0341i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.21 + 2.09i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.46T + 83T^{2} \)
89 \( 1 + (-3.63 + 6.29i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.23T + 97T^{2} \)
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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

−10.98009038121140638970365333484, −9.685982693455542405065185313829, −9.136307630811302790290857654993, −7.68838651775865706601745616671, −7.22769060341325600981622491448, −6.18305212691157326318461745036, −5.50918958540886189228286610358, −4.14110323070457783537543150769, −1.78696425079981958534911999372, −0.51679547235511003298218947746, 2.33358799045613421298476240414, 3.53749833900974091040042168261, 4.64405376846836677555993573994, 5.96806197474045243500397115719, 6.54959343283795841079532592175, 8.379028225815736979159912747438, 9.170967440793557598421622803482, 10.05764615170293195766018385002, 10.70330827786818053825528324202, 11.16357854572745766344532540822

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