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Properties

Label 2-140-28.3-c1-0-0
Degree $2$
Conductor $140$
Sign $-0.994 + 0.102i$
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  + (−1.26 − 0.626i)2-s + (−1.49 + 2.59i)3-s + (1.21 + 1.58i)4-s + (−0.866 + 0.5i)5-s + (3.52 − 2.35i)6-s + (−2.06 − 1.65i)7-s + (−0.546 − 2.77i)8-s + (−2.99 − 5.18i)9-s + (1.41 − 0.0915i)10-s + (−1.93 − 1.11i)11-s + (−5.94 + 0.774i)12-s + 3.17i·13-s + (1.57 + 3.39i)14-s − 2.99i·15-s + (−1.04 + 3.86i)16-s + (−2.98 − 1.72i)17-s + ⋯
L(s)  = 1  + (−0.896 − 0.442i)2-s + (−0.865 + 1.49i)3-s + (0.607 + 0.794i)4-s + (−0.387 + 0.223i)5-s + (1.43 − 0.960i)6-s + (−0.778 − 0.627i)7-s + (−0.193 − 0.981i)8-s + (−0.998 − 1.72i)9-s + (0.446 − 0.0289i)10-s + (−0.584 − 0.337i)11-s + (−1.71 + 0.223i)12-s + 0.879i·13-s + (0.420 + 0.907i)14-s − 0.774i·15-s + (−0.261 + 0.965i)16-s + (−0.723 − 0.417i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.994 + 0.102i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.994 + 0.102i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00765284 - 0.149634i\)
\(L(\frac12)\) \(\approx\) \(0.00765284 - 0.149634i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.626i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.06 + 1.65i)T \)
good3 \( 1 + (1.49 - 2.59i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.93 + 1.11i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.17iT - 13T^{2} \)
17 \( 1 + (2.98 + 1.72i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.02 - 1.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.30 - 1.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.38T + 29T^{2} \)
31 \( 1 + (2.44 - 4.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.59 - 9.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.46iT - 41T^{2} \)
43 \( 1 + 9.95iT - 43T^{2} \)
47 \( 1 + (-3.06 - 5.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.32 - 4.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.55 - 6.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.19 - 1.26i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0456 + 0.0263i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.212iT - 71T^{2} \)
73 \( 1 + (12.8 + 7.43i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.399 + 0.230i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + (-6.07 + 3.51i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.185iT - 97T^{2} \)
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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

−13.54462187034848327959602095121, −12.12969571674329907310539153184, −11.25489617566622851237682194195, −10.59576503400516880962270104005, −9.772049475057443294761962280998, −8.966388499891966365900703980012, −7.35640123044690191893539662311, −6.11218501147341045956209664328, −4.36362226369031652863186534553, −3.32215774306429371542578163452, 0.20797836147086459048649156517, 2.25683339619312794905118083572, 5.45412696717451772903600246428, 6.23425923114206488290601678133, 7.33739754806736776567927203235, 8.062425389919903135704703635283, 9.345971072780355651805879247689, 10.75050869188552129772753874656, 11.59163451437118349736578329423, 12.69057905032482277095778375899

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