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Properties

Label 2-446-223.39-c1-0-5
Degree $2$
Conductor $446$
Sign $0.639 - 0.768i$
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 + (−0.723 + 1.25i)3-s + 4-s + (−0.806 − 1.39i)5-s + (−0.723 + 1.25i)6-s + 0.193·7-s + 8-s + (0.452 + 0.783i)9-s + (−0.806 − 1.39i)10-s + (2.30 + 3.99i)11-s + (−0.723 + 1.25i)12-s + 2.45·13-s + 0.193·14-s + 2.33·15-s + 16-s + 4.36·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.417 + 0.723i)3-s + 0.5·4-s + (−0.360 − 0.624i)5-s + (−0.295 + 0.511i)6-s + 0.0733·7-s + 0.353·8-s + (0.150 + 0.261i)9-s + (−0.255 − 0.441i)10-s + (0.694 + 1.20i)11-s + (−0.208 + 0.361i)12-s + 0.679·13-s + 0.0518·14-s + 0.603·15-s + 0.250·16-s + 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(446\)    =    \(2 \cdot 223\)
Sign: $0.639 - 0.768i$
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: \((2,\ 446,\ (\ :1/2),\ 0.639 - 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70913 + 0.800782i\)
\(L(\frac12)\) \(\approx\) \(1.70913 + 0.800782i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-8.13 + 12.5i)T \)
good3 \( 1 + (0.723 - 1.25i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.806 + 1.39i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 0.193T + 7T^{2} \)
11 \( 1 + (-2.30 - 3.99i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.45T + 13T^{2} \)
17 \( 1 - 4.36T + 17T^{2} \)
19 \( 1 + (2.77 - 4.81i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.515 - 0.892i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.34 + 2.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.54 + 2.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.758 - 1.31i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.69T + 41T^{2} \)
43 \( 1 + (-2.14 + 3.71i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.11 + 7.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.03 - 8.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.81T + 59T^{2} \)
61 \( 1 + (-7.63 + 13.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.17 - 2.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.63 - 2.83i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.10 + 10.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.18 - 2.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.347 - 0.602i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.41 + 4.18i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.53 + 11.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.40098735457763184374884953628, −10.30415060137435503112734537454, −9.762015115256684485423117572752, −8.432021815599058121924539867577, −7.52451283990976761958152899924, −6.29085870894883260941700515957, −5.28083850808441516413068131290, −4.39454633284639217069280582574, −3.73784686161557308361542955635, −1.74558877825490438440912357780, 1.18573452102954480722088648914, 3.05153765112774247797071377425, 3.92388774561365440542662512279, 5.43348817312645140576870220399, 6.41288033959901760800248591822, 6.89750305048962996668517008293, 8.012653314982885035572632177944, 9.081622061347589309019802649980, 10.46124883794828541821697563300, 11.37950017123158069485216800807

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