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Properties

Label 2-140-140.123-c1-0-11
Degree $2$
Conductor $140$
Sign $0.996 + 0.0774i$
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  + (1.04 − 0.952i)2-s + (0.727 + 2.71i)3-s + (0.185 − 1.99i)4-s + (0.178 − 2.22i)5-s + (3.34 + 2.14i)6-s + (0.496 + 2.59i)7-s + (−1.70 − 2.25i)8-s + (−4.24 + 2.44i)9-s + (−1.93 − 2.50i)10-s + (−2.75 − 1.59i)11-s + (5.54 − 0.944i)12-s + (2.41 + 2.41i)13-s + (2.99 + 2.24i)14-s + (6.18 − 1.13i)15-s + (−3.93 − 0.739i)16-s + (−0.600 − 2.24i)17-s + ⋯
L(s)  = 1  + (0.739 − 0.673i)2-s + (0.419 + 1.56i)3-s + (0.0928 − 0.995i)4-s + (0.0799 − 0.996i)5-s + (1.36 + 0.875i)6-s + (0.187 + 0.982i)7-s + (−0.601 − 0.798i)8-s + (−1.41 + 0.816i)9-s + (−0.612 − 0.790i)10-s + (−0.831 − 0.479i)11-s + (1.59 − 0.272i)12-s + (0.670 + 0.670i)13-s + (0.800 + 0.599i)14-s + (1.59 − 0.293i)15-s + (−0.982 − 0.184i)16-s + (−0.145 − 0.543i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.996 + 0.0774i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.996 + 0.0774i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68808 - 0.0654344i\)
\(L(\frac12)\) \(\approx\) \(1.68808 - 0.0654344i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04 + 0.952i)T \)
5 \( 1 + (-0.178 + 2.22i)T \)
7 \( 1 + (-0.496 - 2.59i)T \)
good3 \( 1 + (-0.727 - 2.71i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.75 + 1.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.41 - 2.41i)T + 13iT^{2} \)
17 \( 1 + (0.600 + 2.24i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.39 + 2.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 + 1.39i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 1.72iT - 29T^{2} \)
31 \( 1 + (-3.01 - 1.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.32 - 0.623i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.72T + 41T^{2} \)
43 \( 1 + (-3.96 + 3.96i)T - 43iT^{2} \)
47 \( 1 + (1.63 - 6.10i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-12.6 + 3.37i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.951 - 1.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.83 - 10.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.67 + 1.51i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.562iT - 71T^{2} \)
73 \( 1 + (3.23 - 0.866i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.13 + 7.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.38 + 4.38i)T - 83iT^{2} \)
89 \( 1 + (-2.51 + 1.45i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.9 - 10.9i)T - 97iT^{2} \)
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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

−13.22552167059720155047395610984, −12.02709213093337322669479889257, −11.14439379589092369138525066353, −10.10736053850971963072835352627, −9.134659566832370283525229316397, −8.533036518494843696845826915661, −5.86479019672362797079351449101, −4.97369536140440743084072193051, −4.07199584187886781154055453676, −2.56445638530701183312832970649, 2.34414087006765527717485909203, 3.80434017615142663260431411899, 5.88888039734132142866608747888, 6.75936715702551585972268139788, 7.72731568204979090679868178009, 8.121684984608740876707278218456, 10.28699362712001155699305571516, 11.45810631239986190223419259010, 12.64529922555931870871474021010, 13.34064203474911881837644877094

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