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Properties

Label 2-446-223.183-c1-0-1
Degree $2$
Conductor $446$
Sign $-0.0495 - 0.998i$
Analytic cond. $3.56132$
Root an. cond. $1.88714$
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  + 2-s + (−1.54 − 2.67i)3-s + 4-s + (−2.02 + 3.51i)5-s + (−1.54 − 2.67i)6-s − 0.565·7-s + 8-s + (−3.25 + 5.64i)9-s + (−2.02 + 3.51i)10-s + (−1.91 + 3.32i)11-s + (−1.54 − 2.67i)12-s − 0.971·13-s − 0.565·14-s + 12.5·15-s + 16-s − 2.84·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.890 − 1.54i)3-s + 0.5·4-s + (−0.906 + 1.57i)5-s + (−0.629 − 1.09i)6-s − 0.213·7-s + 0.353·8-s + (−1.08 + 1.88i)9-s + (−0.641 + 1.11i)10-s + (−0.578 + 1.00i)11-s + (−0.445 − 0.771i)12-s − 0.269·13-s − 0.151·14-s + 3.22·15-s + 0.250·16-s − 0.690·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0495 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0495 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(446\)    =    \(2 \cdot 223\)
Sign: $-0.0495 - 0.998i$
Analytic conductor: \(3.56132\)
Root analytic conductor: \(1.88714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{446} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 446,\ (\ :1/2),\ -0.0495 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.464233 + 0.487834i\)
\(L(\frac12)\) \(\approx\) \(0.464233 + 0.487834i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
223 \( 1 + (-1.02 + 14.8i)T \)
good3 \( 1 + (1.54 + 2.67i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.02 - 3.51i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.565T + 7T^{2} \)
11 \( 1 + (1.91 - 3.32i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.971T + 13T^{2} \)
17 \( 1 + 2.84T + 17T^{2} \)
19 \( 1 + (-1.20 - 2.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.79 - 4.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.26 + 2.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.36 + 7.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.19 - 3.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.02T + 41T^{2} \)
43 \( 1 + (-5.81 - 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.96 - 6.86i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.42 - 7.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 + (4.84 + 8.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.617 - 1.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.21 + 12.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.361 - 0.626i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.75 + 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.50 - 7.80i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.32 + 2.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.285 - 0.494i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.36635602663543400733661810729, −11.06316468348818578918013447370, −9.919846926419244232625247129536, −7.66225296652063725745705225711, −7.60990578041888160549833837436, −6.65628519648585311123909790838, −6.02582003043435489665681556293, −4.68503395099401139131505995428, −3.14548745417696059734947965396, −2.06047924215692407867373028697, 0.35595172169669571730223296489, 3.34394658235802406704830986159, 4.23208896944256495498647717309, 5.09376103450895752784141069761, 5.44584325254563453717631873212, 6.90406353959043191171842555326, 8.539987064289788163844103009940, 8.956254676607449589553959523502, 10.23356560514143281502267690637, 11.04514912545354545813864560694

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