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Properties

Label 2-1380-5.4-c3-0-0
Degree $2$
Conductor $1380$
Sign $0.137 + 0.990i$
Analytic cond. $81.4226$
Root an. cond. $9.02344$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + (1.53 + 11.0i)5-s + 33.2i·7-s − 9·9-s − 18.6·11-s − 36.9i·13-s + (−33.2 + 4.59i)15-s − 24.7i·17-s + 21.7·19-s − 99.6·21-s − 23i·23-s + (−120. + 33.9i)25-s − 27i·27-s − 168.·29-s − 26.5·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.137 + 0.990i)5-s + 1.79i·7-s − 0.333·9-s − 0.511·11-s − 0.788i·13-s + (−0.571 + 0.0791i)15-s − 0.352i·17-s + 0.262·19-s − 1.03·21-s − 0.208i·23-s + (−0.962 + 0.271i)25-s − 0.192i·27-s − 1.08·29-s − 0.154·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.137 + 0.990i$
Analytic conductor: \(81.4226\)
Root analytic conductor: \(9.02344\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :3/2),\ 0.137 + 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.09810367795\)
\(L(\frac12)\) \(\approx\) \(0.09810367795\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
5 \( 1 + (-1.53 - 11.0i)T \)
23 \( 1 + 23iT \)
good7 \( 1 - 33.2iT - 343T^{2} \)
11 \( 1 + 18.6T + 1.33e3T^{2} \)
13 \( 1 + 36.9iT - 2.19e3T^{2} \)
17 \( 1 + 24.7iT - 4.91e3T^{2} \)
19 \( 1 - 21.7T + 6.85e3T^{2} \)
29 \( 1 + 168.T + 2.43e4T^{2} \)
31 \( 1 + 26.5T + 2.97e4T^{2} \)
37 \( 1 + 286. iT - 5.06e4T^{2} \)
41 \( 1 + 137.T + 6.89e4T^{2} \)
43 \( 1 - 117. iT - 7.95e4T^{2} \)
47 \( 1 + 43.6iT - 1.03e5T^{2} \)
53 \( 1 - 15.9iT - 1.48e5T^{2} \)
59 \( 1 + 345.T + 2.05e5T^{2} \)
61 \( 1 + 30.8T + 2.26e5T^{2} \)
67 \( 1 + 623. iT - 3.00e5T^{2} \)
71 \( 1 - 454.T + 3.57e5T^{2} \)
73 \( 1 - 409. iT - 3.89e5T^{2} \)
79 \( 1 - 625.T + 4.93e5T^{2} \)
83 \( 1 - 574. iT - 5.71e5T^{2} \)
89 \( 1 - 579.T + 7.04e5T^{2} \)
97 \( 1 - 621. iT - 9.12e5T^{2} \)
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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

−9.691521798154240483985671092066, −9.218110247523694010464320556706, −8.264463742327828619634682073829, −7.50823264914527010105188323023, −6.35368859451188827865953697530, −5.61616568582118428235701261103, −5.09173978481953004192648844315, −3.60854088267441519059642004569, −2.80234234111100931048370608862, −2.11180795304637057655199278439, 0.02317293332280311417897727491, 1.06368961950559233053747391842, 1.86217771167332470307812950063, 3.46588732678592896042554427957, 4.32511877200219188618736628916, 5.12314141733414227058925482703, 6.20687651103164129047967403202, 7.10458461811163285876687523630, 7.71041015956690828697459058327, 8.453159459194371075084302745986

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