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Properties

Label 2-1440-40.3-c1-0-6
Degree $2$
Conductor $1440$
Sign $-0.0346 - 0.999i$
Analytic cond. $11.4984$
Root an. cond. $3.39093$
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  + (1.83 + 1.28i)5-s + (−2.94 − 2.94i)7-s + 1.61·11-s + (−2.50 + 2.50i)13-s + (−4.59 + 4.59i)17-s + 4i·19-s + (−1.09 + 1.09i)23-s + (1.70 + 4.70i)25-s + 4.75·29-s + 5.01i·31-s + (−1.61 − 9.18i)35-s + (2.50 + 2.50i)37-s + 9.18·41-s + (7.40 + 7.40i)43-s + (−7.32 − 7.32i)47-s + ⋯
L(s)  = 1  + (0.818 + 0.574i)5-s + (−1.11 − 1.11i)7-s + 0.485·11-s + (−0.696 + 0.696i)13-s + (−1.11 + 1.11i)17-s + 0.917i·19-s + (−0.227 + 0.227i)23-s + (0.340 + 0.940i)25-s + 0.882·29-s + 0.901i·31-s + (−0.272 − 1.55i)35-s + (0.412 + 0.412i)37-s + 1.43·41-s + (1.12 + 1.12i)43-s + (−1.06 − 1.06i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0346 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0346 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.0346 - 0.999i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.0346 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.243933765\)
\(L(\frac12)\) \(\approx\) \(1.243933765\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-1.83 - 1.28i)T \)
good7 \( 1 + (2.94 + 2.94i)T + 7iT^{2} \)
11 \( 1 - 1.61T + 11T^{2} \)
13 \( 1 + (2.50 - 2.50i)T - 13iT^{2} \)
17 \( 1 + (4.59 - 4.59i)T - 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (1.09 - 1.09i)T - 23iT^{2} \)
29 \( 1 - 4.75T + 29T^{2} \)
31 \( 1 - 5.01iT - 31T^{2} \)
37 \( 1 + (-2.50 - 2.50i)T + 37iT^{2} \)
41 \( 1 - 9.18T + 41T^{2} \)
43 \( 1 + (-7.40 - 7.40i)T + 43iT^{2} \)
47 \( 1 + (7.32 + 7.32i)T + 47iT^{2} \)
53 \( 1 + (3.11 - 3.11i)T - 53iT^{2} \)
59 \( 1 - 1.61iT - 59T^{2} \)
61 \( 1 + 6.78iT - 61T^{2} \)
67 \( 1 + (7.40 - 7.40i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5 - 5i)T + 73iT^{2} \)
79 \( 1 + 5.01T + 79T^{2} \)
83 \( 1 + (-7.57 - 7.57i)T + 83iT^{2} \)
89 \( 1 + 2.74iT - 89T^{2} \)
97 \( 1 + (-2.40 + 2.40i)T - 97iT^{2} \)
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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

−9.739401316967123690728768091558, −9.236789474580606452438411838110, −8.078171568164898262356765980948, −7.00385054929279452079969567047, −6.56140718399120817908881977375, −5.92531142534647739901492029979, −4.49857314056135222563230948124, −3.74877603506861582156680493670, −2.68011965941054049729612715348, −1.46074903352654059630065861504, 0.48949394098166491179345700631, 2.35699216293911978513610636034, 2.81029302353963738698362188492, 4.39487104958987285614332501829, 5.22175863218761133339740584121, 6.08410002695418494323237957747, 6.63579280710943064059217336539, 7.73474586287730207990282639013, 8.978660545250290866370213585614, 9.202631936516771189710237914309

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