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Properties

Label 2-1332-37.10-c1-0-5
Degree $2$
Conductor $1332$
Sign $0.999 + 0.0330i$
Analytic cond. $10.6360$
Root an. cond. $3.26129$
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  + (−0.5 − 0.866i)5-s + (−0.370 − 0.641i)7-s + 2.91·11-s + (1.54 + 2.68i)13-s + (−3.41 + 5.92i)17-s + (−0.0887 − 0.153i)19-s + 3.43·23-s + (2 − 3.46i)25-s + 4.61·29-s + 2.91·31-s + (−0.370 + 0.641i)35-s + (−2.04 − 5.72i)37-s + (0.322 + 0.558i)41-s − 3.48·43-s + 9.79·47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.140 − 0.242i)7-s + 0.879·11-s + (0.429 + 0.743i)13-s + (−0.829 + 1.43i)17-s + (−0.0203 − 0.0352i)19-s + 0.716·23-s + (0.400 − 0.692i)25-s + 0.856·29-s + 0.524·31-s + (−0.0626 + 0.108i)35-s + (−0.336 − 0.941i)37-s + (0.0503 + 0.0872i)41-s − 0.531·43-s + 1.42·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1332\)    =    \(2^{2} \cdot 3^{2} \cdot 37\)
Sign: $0.999 + 0.0330i$
Analytic conductor: \(10.6360\)
Root analytic conductor: \(3.26129\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1332} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1332,\ (\ :1/2),\ 0.999 + 0.0330i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.663386520\)
\(L(\frac12)\) \(\approx\) \(1.663386520\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( 1 + (2.04 + 5.72i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.370 + 0.641i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.91T + 11T^{2} \)
13 \( 1 + (-1.54 - 2.68i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.41 - 5.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0887 + 0.153i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.43T + 23T^{2} \)
29 \( 1 - 4.61T + 29T^{2} \)
31 \( 1 - 2.91T + 31T^{2} \)
41 \( 1 + (-0.322 - 0.558i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 3.48T + 43T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 + (-3.37 + 5.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.72 + 6.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.67 - 6.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.18 - 8.98i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.72 - 4.72i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + (-6.26 - 10.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.18 - 7.24i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.85 + 13.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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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

−9.527271733253261540086852075822, −8.641695061469422082061427465308, −8.335782609924534116265758374010, −6.86519848418201452132673886181, −6.58847258605684899322622688104, −5.42448333657631227873613331821, −4.24633070922421771709671459338, −3.83074079255355082464635778333, −2.28741177021536923767863479795, −1.01722322989816621173880096772, 0.951339968884921122449917374715, 2.59015646462567796505421170357, 3.39219137399972318580627549458, 4.52907070956754543875066719200, 5.41693757723104305876692832523, 6.53165663207817357160863731683, 7.01579540670176613956687571846, 8.028264658798241861523753950173, 8.944594856691372945859188693723, 9.458883376647994548606176229091

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