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Properties

Label 2-1380-115.22-c1-0-21
Degree $2$
Conductor $1380$
Sign $-0.350 + 0.936i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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  + (0.707 − 0.707i)3-s + (0.577 − 2.16i)5-s + (0.575 − 0.575i)7-s − 1.00i·9-s − 2.55i·11-s + (−1.98 + 1.98i)13-s + (−1.11 − 1.93i)15-s + (1.42 − 1.42i)17-s + 3.58·19-s − 0.813i·21-s + (4.75 − 0.644i)23-s + (−4.33 − 2.49i)25-s + (−0.707 − 0.707i)27-s − 3.90i·29-s − 7.98·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.258 − 0.966i)5-s + (0.217 − 0.217i)7-s − 0.333i·9-s − 0.771i·11-s + (−0.549 + 0.549i)13-s + (−0.289 − 0.499i)15-s + (0.344 − 0.344i)17-s + 0.823·19-s − 0.177i·21-s + (0.990 − 0.134i)23-s + (−0.866 − 0.498i)25-s + (−0.136 − 0.136i)27-s − 0.725i·29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.350 + 0.936i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -0.350 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.880726224\)
\(L(\frac12)\) \(\approx\) \(1.880726224\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.577 + 2.16i)T \)
23 \( 1 + (-4.75 + 0.644i)T \)
good7 \( 1 + (-0.575 + 0.575i)T - 7iT^{2} \)
11 \( 1 + 2.55iT - 11T^{2} \)
13 \( 1 + (1.98 - 1.98i)T - 13iT^{2} \)
17 \( 1 + (-1.42 + 1.42i)T - 17iT^{2} \)
19 \( 1 - 3.58T + 19T^{2} \)
29 \( 1 + 3.90iT - 29T^{2} \)
31 \( 1 + 7.98T + 31T^{2} \)
37 \( 1 + (-5.07 + 5.07i)T - 37iT^{2} \)
41 \( 1 + 4.13T + 41T^{2} \)
43 \( 1 + (5.16 + 5.16i)T + 43iT^{2} \)
47 \( 1 + (0.115 + 0.115i)T + 47iT^{2} \)
53 \( 1 + (-2.73 - 2.73i)T + 53iT^{2} \)
59 \( 1 - 4.87iT - 59T^{2} \)
61 \( 1 + 5.54iT - 61T^{2} \)
67 \( 1 + (5.83 - 5.83i)T - 67iT^{2} \)
71 \( 1 - 0.976T + 71T^{2} \)
73 \( 1 + (0.307 - 0.307i)T - 73iT^{2} \)
79 \( 1 - 6.52T + 79T^{2} \)
83 \( 1 + (4.68 + 4.68i)T + 83iT^{2} \)
89 \( 1 - 3.06T + 89T^{2} \)
97 \( 1 + (-5.38 + 5.38i)T - 97iT^{2} \)
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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

−9.207045081549451197753933370315, −8.634180854410582472657686024133, −7.70900509387412805158091292182, −7.09665393010982863886267157149, −5.91044859182889475185681839691, −5.18730746675873347051357081532, −4.21496941559931422527181273724, −3.10708484765534806864288967180, −1.88147583997962565891989258582, −0.73577126209534656358884572929, 1.74982568352179476326767530595, 2.87085852827650026531298939880, 3.56074144857781054050153044252, 4.87422058349900288561754858172, 5.52870092888607220434638962367, 6.73096468667161518363744961096, 7.39876207517033287191757151153, 8.137131091074146251168192358522, 9.233629775828746211076298609031, 9.823785028349113184818168287197

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