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Properties

Label 2-140-5.3-c2-0-1
Degree $2$
Conductor $140$
Sign $-0.00402 - 0.999i$
Analytic cond. $3.81472$
Root an. cond. $1.95313$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.87 + 2.87i)3-s + (−1.12 + 4.87i)5-s + (−1.87 + 1.87i)7-s + 7.48i·9-s − 3.51·11-s + (0.612 + 0.612i)13-s + (−17.2 + 10.7i)15-s + (14.0 − 14.0i)17-s − 2.25i·19-s − 10.7·21-s + (29.7 + 29.7i)23-s + (−22.4 − 11i)25-s + (4.35 − 4.35i)27-s + 16.9i·29-s + 33.6·31-s + ⋯
L(s)  = 1  + (0.956 + 0.956i)3-s + (−0.225 + 0.974i)5-s + (−0.267 + 0.267i)7-s + 0.831i·9-s − 0.319·11-s + (0.0471 + 0.0471i)13-s + (−1.14 + 0.716i)15-s + (0.829 − 0.829i)17-s − 0.118i·19-s − 0.511·21-s + (1.29 + 1.29i)23-s + (−0.897 − 0.440i)25-s + (0.161 − 0.161i)27-s + 0.583i·29-s + 1.08·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00402 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00402 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.00402 - 0.999i$
Analytic conductor: \(3.81472\)
Root analytic conductor: \(1.95313\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1),\ -0.00402 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.24012 + 1.24512i\)
\(L(\frac12)\) \(\approx\) \(1.24012 + 1.24512i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.12 - 4.87i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (-2.87 - 2.87i)T + 9iT^{2} \)
11 \( 1 + 3.51T + 121T^{2} \)
13 \( 1 + (-0.612 - 0.612i)T + 169iT^{2} \)
17 \( 1 + (-14.0 + 14.0i)T - 289iT^{2} \)
19 \( 1 + 2.25iT - 361T^{2} \)
23 \( 1 + (-29.7 - 29.7i)T + 529iT^{2} \)
29 \( 1 - 16.9iT - 841T^{2} \)
31 \( 1 - 33.6T + 961T^{2} \)
37 \( 1 + (-14.7 + 14.7i)T - 1.36e3iT^{2} \)
41 \( 1 + 5.67T + 1.68e3T^{2} \)
43 \( 1 + (38.4 + 38.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-49.7 + 49.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (53.7 + 53.7i)T + 2.80e3iT^{2} \)
59 \( 1 + 39.2iT - 3.48e3T^{2} \)
61 \( 1 + 98.9T + 3.72e3T^{2} \)
67 \( 1 + (31.2 - 31.2i)T - 4.48e3iT^{2} \)
71 \( 1 - 138.T + 5.04e3T^{2} \)
73 \( 1 + (79.4 + 79.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 151. iT - 6.24e3T^{2} \)
83 \( 1 + (-42.9 - 42.9i)T + 6.88e3iT^{2} \)
89 \( 1 - 117. iT - 7.92e3T^{2} \)
97 \( 1 + (-42.3 + 42.3i)T - 9.40e3iT^{2} \)
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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

−13.53074992032866593191598935879, −12.03503377997953400135375220392, −10.93495208179109031921618455046, −9.950574115304894583623633294533, −9.216267459155911135472422620315, −7.988985703782332411869011192405, −6.85993783828339847998083874660, −5.20173329403313459028962709074, −3.60319312592632309477094590812, −2.79975408087635323133465310738, 1.20445809648615046433197752184, 2.94611465659381743290186764322, 4.59312966598415261853118287766, 6.24984255587538263771242590992, 7.63650909990148857417987452592, 8.277286199517623550927759109312, 9.241543771850162591876862866448, 10.54922285174896364357859481865, 12.09286992869301511366718099166, 12.83571339247846915838923121971

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