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Properties

Label 2-444-12.11-c1-0-56
Degree $2$
Conductor $444$
Sign $0.993 + 0.117i$
Analytic cond. $3.54535$
Root an. cond. $1.88291$
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  + (1.40 + 0.163i)2-s + (1.72 − 0.196i)3-s + (1.94 + 0.458i)4-s − 3.43i·5-s + (2.44 + 0.00483i)6-s + 4.57i·7-s + (2.65 + 0.962i)8-s + (2.92 − 0.676i)9-s + (0.560 − 4.82i)10-s − 3.04·11-s + (3.44 + 0.406i)12-s − 6.03·13-s + (−0.746 + 6.42i)14-s + (−0.675 − 5.91i)15-s + (3.57 + 1.78i)16-s + 0.831i·17-s + ⋯
L(s)  = 1  + (0.993 + 0.115i)2-s + (0.993 − 0.113i)3-s + (0.973 + 0.229i)4-s − 1.53i·5-s + (0.999 + 0.00197i)6-s + 1.72i·7-s + (0.940 + 0.340i)8-s + (0.974 − 0.225i)9-s + (0.177 − 1.52i)10-s − 0.916·11-s + (0.993 + 0.117i)12-s − 1.67·13-s + (−0.199 + 1.71i)14-s + (−0.174 − 1.52i)15-s + (0.894 + 0.446i)16-s + 0.201i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $0.993 + 0.117i$
Analytic conductor: \(3.54535\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :1/2),\ 0.993 + 0.117i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.24190 - 0.190942i\)
\(L(\frac12)\) \(\approx\) \(3.24190 - 0.190942i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.163i)T \)
3 \( 1 + (-1.72 + 0.196i)T \)
37 \( 1 - T \)
good5 \( 1 + 3.43iT - 5T^{2} \)
7 \( 1 - 4.57iT - 7T^{2} \)
11 \( 1 + 3.04T + 11T^{2} \)
13 \( 1 + 6.03T + 13T^{2} \)
17 \( 1 - 0.831iT - 17T^{2} \)
19 \( 1 + 2.65iT - 19T^{2} \)
23 \( 1 + 1.78T + 23T^{2} \)
29 \( 1 + 7.42iT - 29T^{2} \)
31 \( 1 + 1.01iT - 31T^{2} \)
41 \( 1 - 5.34iT - 41T^{2} \)
43 \( 1 - 8.74iT - 43T^{2} \)
47 \( 1 + 1.38T + 47T^{2} \)
53 \( 1 - 6.20iT - 53T^{2} \)
59 \( 1 - 8.65T + 59T^{2} \)
61 \( 1 + 6.78T + 61T^{2} \)
67 \( 1 + 10.8iT - 67T^{2} \)
71 \( 1 + 1.12T + 71T^{2} \)
73 \( 1 - 5.61T + 73T^{2} \)
79 \( 1 - 4.16iT - 79T^{2} \)
83 \( 1 - 1.49T + 83T^{2} \)
89 \( 1 + 15.3iT - 89T^{2} \)
97 \( 1 - 16.7T + 97T^{2} \)
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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

−11.62988211216116149128231238774, −9.914483423865971462151549667353, −9.200733945399418899006341342247, −8.212921910099423820978266740076, −7.67543235936068400370063936642, −6.11386806134987244087345127980, −5.07373466344498467023821773463, −4.54766231557139148928805757891, −2.79817173769608782411772022679, −2.09093025389087749882372815584, 2.20200304531778371946606728673, 3.17468270462936497983140100507, 3.96775490782634623698839394995, 5.12152313931451520643856861197, 6.88124019815175423874903027469, 7.20498688628531516052816557375, 7.86927122266192967971625556849, 9.886014997657404313663319514223, 10.37012789631914574502472415462, 10.82622058390954410210129920796

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