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Properties

Label 2-446-223.39-c1-0-6
Degree $2$
Conductor $446$
Sign $0.972 - 0.234i$
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 + (0.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + 4·7-s − 8-s + (1 + 1.73i)9-s + (−0.5 − 0.866i)10-s + (1.5 + 2.59i)11-s + (0.5 − 0.866i)12-s − 6·13-s − 4·14-s + 0.999·15-s + 16-s + 2·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.288 − 0.499i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (−0.204 + 0.353i)6-s + 1.51·7-s − 0.353·8-s + (0.333 + 0.577i)9-s + (−0.158 − 0.273i)10-s + (0.452 + 0.783i)11-s + (0.144 − 0.249i)12-s − 1.66·13-s − 1.06·14-s + 0.258·15-s + 0.250·16-s + 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(446\)    =    \(2 \cdot 223\)
Sign: $0.972 - 0.234i$
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: \((2,\ 446,\ (\ :1/2),\ 0.972 - 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32903 + 0.158184i\)
\(L(\frac12)\) \(\approx\) \(1.32903 + 0.158184i\)
\(L(\frac{3}{2})\) 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 \( 1 + (14 - 5.19i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + (-4.5 + 7.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.5 - 4.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.5 + 7.79i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.5 + 2.59i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.87674307827051269127210149097, −10.30804846636960323366633224926, −9.330902531358128447905348487320, −8.218615588977790226278756074337, −7.51106906973175091634153742766, −7.00525067486069476568448606628, −5.42988925606068373484747932559, −4.41090381433194355574196239511, −2.40707935905835771923030077474, −1.66488588265969981710572083852, 1.20512999023140444990448211418, 2.72867424841901022524385981552, 4.34675352525955637728884807942, 5.17368611403784948619434279212, 6.56188425785692976055843762617, 7.68309656228203128479833481314, 8.469857521694126426972111050148, 9.262741533766614024679382652318, 10.00243992120207769428364309038, 10.94785906528394400422863814662

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