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Properties

Label 2-446-223.39-c1-0-14
Degree $2$
Conductor $446$
Sign $0.268 + 0.963i$
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 + (0.686 − 1.18i)3-s + 4-s + (−1.36 − 2.36i)5-s + (0.686 − 1.18i)6-s + 2.10·7-s + 8-s + (0.556 + 0.964i)9-s + (−1.36 − 2.36i)10-s + (−2.22 − 3.84i)11-s + (0.686 − 1.18i)12-s − 2.07·13-s + 2.10·14-s − 3.75·15-s + 16-s + 3.23·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.396 − 0.686i)3-s + 0.5·4-s + (−0.611 − 1.05i)5-s + (0.280 − 0.485i)6-s + 0.796·7-s + 0.353·8-s + (0.185 + 0.321i)9-s + (−0.432 − 0.748i)10-s + (−0.669 − 1.15i)11-s + (0.198 − 0.343i)12-s − 0.575·13-s + 0.563·14-s − 0.969·15-s + 0.250·16-s + 0.783·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(446\)    =    \(2 \cdot 223\)
Sign: $0.268 + 0.963i$
Analytic conductor: \(3.56132\)
Root analytic conductor: \(1.88714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{446} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 446,\ (\ :1/2),\ 0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81226 - 1.37616i\)
\(L(\frac12)\) \(\approx\) \(1.81226 - 1.37616i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.68 - 13.8i)T \)
good3 \( 1 + (-0.686 + 1.18i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.36 + 2.36i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 2.10T + 7T^{2} \)
11 \( 1 + (2.22 + 3.84i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.07T + 13T^{2} \)
17 \( 1 - 3.23T + 17T^{2} \)
19 \( 1 + (3.16 - 5.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.34 + 2.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.242 - 0.419i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.01 + 1.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.05 + 1.82i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.71T + 41T^{2} \)
43 \( 1 + (4.86 - 8.41i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.04 - 8.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.76 - 4.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.73T + 59T^{2} \)
61 \( 1 + (2.54 - 4.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.51 - 4.35i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.293 - 0.508i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.67 - 4.62i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.51 + 13.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.72 + 11.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.67 - 13.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.46 - 11.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.14177418810001307304030087936, −10.23881988361161908904466731393, −8.658779997665097590390537555123, −8.032739271647126101349530197280, −7.53523602257639497311526679459, −6.01577278969442387218875786428, −5.04354058865445802507977248904, −4.18462583465857475065564905034, −2.70516627340991308170697601187, −1.26374317470044882077244338821, 2.34462898120143514893716456652, 3.41260067063660278001353285088, 4.43742574200985000940153536739, 5.21933684709343775482851966049, 6.90086866968906978392413416651, 7.32484989398196263304306629864, 8.477984907954502456589659616570, 9.790245691968217468651283293153, 10.45080790074299861133526458306, 11.28517836976435529326734520713

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