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Properties

Label 2-140-140.107-c1-0-4
Degree $2$
Conductor $140$
Sign $-0.235 - 0.971i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
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  + (0.0366 + 1.41i)2-s + (0.107 − 0.402i)3-s + (−1.99 + 0.103i)4-s + (1.27 + 1.83i)5-s + (0.572 + 0.137i)6-s + (−1.30 + 2.30i)7-s + (−0.219 − 2.81i)8-s + (2.44 + 1.41i)9-s + (−2.54 + 1.87i)10-s + (−0.725 + 0.418i)11-s + (−0.173 + 0.814i)12-s + (−1.16 + 1.16i)13-s + (−3.30 − 1.75i)14-s + (0.875 − 0.316i)15-s + (3.97 − 0.414i)16-s + (1.32 − 4.94i)17-s + ⋯
L(s)  = 1  + (0.0259 + 0.999i)2-s + (0.0622 − 0.232i)3-s + (−0.998 + 0.0518i)4-s + (0.571 + 0.820i)5-s + (0.233 + 0.0561i)6-s + (−0.492 + 0.870i)7-s + (−0.0777 − 0.996i)8-s + (0.815 + 0.471i)9-s + (−0.805 + 0.592i)10-s + (−0.218 + 0.126i)11-s + (−0.0500 + 0.235i)12-s + (−0.322 + 0.322i)13-s + (−0.882 − 0.469i)14-s + (0.226 − 0.0815i)15-s + (0.994 − 0.103i)16-s + (0.321 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.235 - 0.971i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.235 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.672831 + 0.855723i\)
\(L(\frac12)\) \(\approx\) \(0.672831 + 0.855723i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0366 - 1.41i)T \)
5 \( 1 + (-1.27 - 1.83i)T \)
7 \( 1 + (1.30 - 2.30i)T \)
good3 \( 1 + (-0.107 + 0.402i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.725 - 0.418i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.16 - 1.16i)T - 13iT^{2} \)
17 \( 1 + (-1.32 + 4.94i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.91 + 5.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.34 - 0.896i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.00iT - 29T^{2} \)
31 \( 1 + (-7.03 + 4.06i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.711 + 0.190i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.0958T + 41T^{2} \)
43 \( 1 + (-4.87 - 4.87i)T + 43iT^{2} \)
47 \( 1 + (-1.41 - 5.28i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (12.8 + 3.43i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.46 + 7.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.919 + 1.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.515 + 0.138i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 13.6iT - 71T^{2} \)
73 \( 1 + (5.78 + 1.55i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.30 - 9.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.36 + 4.36i)T + 83iT^{2} \)
89 \( 1 + (-2.50 - 1.44i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.24 - 4.24i)T + 97iT^{2} \)
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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

−13.64192742185501804166561340298, −12.80600743722344221670443531532, −11.53004313829458272981303070138, −9.835032184782605228260987320218, −9.491306055728013007860463576144, −7.85672065636154291586341122812, −6.97051045086009369789695601130, −5.99897531194603067057658191220, −4.75014020005056211847555816405, −2.72104953339073150533551713316, 1.34826230270172419826486134523, 3.50458634576405011294765392761, 4.59878499346264347817493671731, 6.00701451112173033253879249453, 7.82868288491734327129770019082, 9.041436695246302838549400932076, 10.23058353982402489968838104063, 10.27967144737428473048315881149, 12.20442033187702367594509631640, 12.65744072828612703286352794233

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