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Properties

Label 2-140-20.7-c1-0-10
Degree $2$
Conductor $140$
Sign $0.966 + 0.256i$
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  + (0.802 − 1.16i)2-s + (1.75 + 1.75i)3-s + (−0.713 − 1.86i)4-s + (−0.854 + 2.06i)5-s + (3.45 − 0.637i)6-s + (0.707 − 0.707i)7-s + (−2.74 − 0.667i)8-s + 3.17i·9-s + (1.72 + 2.65i)10-s − 5.34i·11-s + (2.02 − 4.53i)12-s + (−2.61 + 2.61i)13-s + (−0.256 − 1.39i)14-s + (−5.13 + 2.12i)15-s + (−2.98 + 2.66i)16-s + (−1.04 − 1.04i)17-s + ⋯
L(s)  = 1  + (0.567 − 0.823i)2-s + (1.01 + 1.01i)3-s + (−0.356 − 0.934i)4-s + (−0.382 + 0.924i)5-s + (1.41 − 0.260i)6-s + (0.267 − 0.267i)7-s + (−0.971 − 0.235i)8-s + 1.05i·9-s + (0.544 + 0.838i)10-s − 1.61i·11-s + (0.585 − 1.30i)12-s + (−0.726 + 0.726i)13-s + (−0.0685 − 0.371i)14-s + (−1.32 + 0.549i)15-s + (−0.745 + 0.666i)16-s + (−0.252 − 0.252i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.966 + 0.256i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.966 + 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67056 - 0.217721i\)
\(L(\frac12)\) \(\approx\) \(1.67056 - 0.217721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.802 + 1.16i)T \)
5 \( 1 + (0.854 - 2.06i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-1.75 - 1.75i)T + 3iT^{2} \)
11 \( 1 + 5.34iT - 11T^{2} \)
13 \( 1 + (2.61 - 2.61i)T - 13iT^{2} \)
17 \( 1 + (1.04 + 1.04i)T + 17iT^{2} \)
19 \( 1 - 0.260T + 19T^{2} \)
23 \( 1 + (1.06 + 1.06i)T + 23iT^{2} \)
29 \( 1 - 1.36iT - 29T^{2} \)
31 \( 1 - 2.05iT - 31T^{2} \)
37 \( 1 + (-8.20 - 8.20i)T + 37iT^{2} \)
41 \( 1 + 9.70T + 41T^{2} \)
43 \( 1 + (-6.86 - 6.86i)T + 43iT^{2} \)
47 \( 1 + (0.976 - 0.976i)T - 47iT^{2} \)
53 \( 1 + (-1.81 + 1.81i)T - 53iT^{2} \)
59 \( 1 - 4.53T + 59T^{2} \)
61 \( 1 + 4.13T + 61T^{2} \)
67 \( 1 + (-4.96 + 4.96i)T - 67iT^{2} \)
71 \( 1 + 7.29iT - 71T^{2} \)
73 \( 1 + (-9.65 + 9.65i)T - 73iT^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + (5.83 + 5.83i)T + 83iT^{2} \)
89 \( 1 - 7.63iT - 89T^{2} \)
97 \( 1 + (-11.5 - 11.5i)T + 97iT^{2} \)
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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

−13.49318442571797482535241568052, −11.81341650376145477910551698737, −11.04548798573902537149514961499, −10.19228097603908030204434889079, −9.241272201529511114862921659715, −8.148391730461915968889175150740, −6.42416983439314052664603630995, −4.70370864801492065024952501213, −3.60518448777492865011752982162, −2.72811598391749997763021257374, 2.31360383375663034777047407869, 4.17164611473593928093120008922, 5.40201295584267772019312101880, 7.11746345998796472829467447911, 7.73890966589229686724420902766, 8.572270343195427800479981143913, 9.613225265771678876263203179247, 11.87080885531998647183700839778, 12.67993977242720023375833571139, 13.03917927752351688774686304211

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