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Properties

Label 2-140-140.107-c1-0-14
Degree $2$
Conductor $140$
Sign $0.521 + 0.853i$
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  + (0.942 − 1.05i)2-s + (−0.322 + 1.20i)3-s + (−0.223 − 1.98i)4-s + (−0.381 − 2.20i)5-s + (0.965 + 1.47i)6-s + (2.60 − 0.451i)7-s + (−2.30 − 1.63i)8-s + (1.25 + 0.722i)9-s + (−2.68 − 1.67i)10-s + (0.824 − 0.476i)11-s + (2.46 + 0.372i)12-s + (−3.15 + 3.15i)13-s + (1.98 − 3.17i)14-s + (2.77 + 0.251i)15-s + (−3.90 + 0.886i)16-s + (−0.688 + 2.56i)17-s + ⋯
L(s)  = 1  + (0.666 − 0.745i)2-s + (−0.186 + 0.695i)3-s + (−0.111 − 0.993i)4-s + (−0.170 − 0.985i)5-s + (0.394 + 0.602i)6-s + (0.985 − 0.170i)7-s + (−0.815 − 0.579i)8-s + (0.417 + 0.240i)9-s + (−0.848 − 0.529i)10-s + (0.248 − 0.143i)11-s + (0.711 + 0.107i)12-s + (−0.875 + 0.875i)13-s + (0.529 − 0.848i)14-s + (0.716 + 0.0649i)15-s + (−0.975 + 0.221i)16-s + (−0.166 + 0.623i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 + 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.521 + 0.853i$
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: \((2,\ 140,\ (\ :1/2),\ 0.521 + 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26697 - 0.710923i\)
\(L(\frac12)\) \(\approx\) \(1.26697 - 0.710923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.942 + 1.05i)T \)
5 \( 1 + (0.381 + 2.20i)T \)
7 \( 1 + (-2.60 + 0.451i)T \)
good3 \( 1 + (0.322 - 1.20i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.824 + 0.476i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.15 - 3.15i)T - 13iT^{2} \)
17 \( 1 + (0.688 - 2.56i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.99 - 3.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.528 - 0.141i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 7.23iT - 29T^{2} \)
31 \( 1 + (-3.41 + 1.97i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.00 - 0.269i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.71T + 41T^{2} \)
43 \( 1 + (-8.73 - 8.73i)T + 43iT^{2} \)
47 \( 1 + (-0.830 - 3.09i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.83 - 0.760i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (6.30 + 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.01 + 5.21i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (14.3 + 3.85i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.60iT - 71T^{2} \)
73 \( 1 + (14.2 + 3.80i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.34 - 9.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.72 + 5.72i)T + 83iT^{2} \)
89 \( 1 + (-1.83 - 1.05i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.08 + 3.08i)T + 97iT^{2} \)
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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

−12.84204000291424395649800859970, −11.94437029293370057797019122409, −11.12882321441716205093087844289, −10.06401324086200454613149646359, −9.181163698226873895304034859719, −7.81872620935404460717327165591, −5.91220681883399855035256696296, −4.54664024576756491092347946676, −4.27102755881285448650985707059, −1.77763940322548329785701910621, 2.64732285800970374544578323199, 4.40399366429766928047806739727, 5.73470926946425856749186625336, 7.11986717125233702321477773134, 7.39659849678677047590349480000, 8.795587392043891557460220225708, 10.47957921986113896485302250759, 11.68800092780276873578663183309, 12.35391884213217998083133464504, 13.44285518173385726366608707866

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