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Properties

Label 2-140-140.123-c1-0-17
Degree $2$
Conductor $140$
Sign $-0.409 + 0.912i$
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  + (0.839 − 1.13i)2-s + (−0.290 − 1.08i)3-s + (−0.592 − 1.91i)4-s + (−2.11 + 0.727i)5-s + (−1.47 − 0.578i)6-s + (2.16 − 1.51i)7-s + (−2.67 − 0.928i)8-s + (1.50 − 0.871i)9-s + (−0.946 + 3.01i)10-s + (−2.58 − 1.49i)11-s + (−1.89 + 1.19i)12-s + (4.05 + 4.05i)13-s + (0.0911 − 3.74i)14-s + (1.40 + 2.07i)15-s + (−3.29 + 2.26i)16-s + (0.617 + 2.30i)17-s + ⋯
L(s)  = 1  + (0.593 − 0.805i)2-s + (−0.167 − 0.625i)3-s + (−0.296 − 0.955i)4-s + (−0.945 + 0.325i)5-s + (−0.602 − 0.236i)6-s + (0.819 − 0.573i)7-s + (−0.944 − 0.328i)8-s + (0.503 − 0.290i)9-s + (−0.299 + 0.954i)10-s + (−0.779 − 0.449i)11-s + (−0.547 + 0.345i)12-s + (1.12 + 1.12i)13-s + (0.0243 − 0.999i)14-s + (0.361 + 0.536i)15-s + (−0.824 + 0.565i)16-s + (0.149 + 0.558i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.409 + 0.912i$
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: \((2,\ 140,\ (\ :1/2),\ -0.409 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.687885 - 1.06234i\)
\(L(\frac12)\) \(\approx\) \(0.687885 - 1.06234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.839 + 1.13i)T \)
5 \( 1 + (2.11 - 0.727i)T \)
7 \( 1 + (-2.16 + 1.51i)T \)
good3 \( 1 + (0.290 + 1.08i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.58 + 1.49i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.05 - 4.05i)T + 13iT^{2} \)
17 \( 1 + (-0.617 - 2.30i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.25 - 2.17i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.79 - 1.55i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.55iT - 29T^{2} \)
31 \( 1 + (3.12 + 1.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.61 - 1.23i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.93T + 41T^{2} \)
43 \( 1 + (7.62 - 7.62i)T - 43iT^{2} \)
47 \( 1 + (0.765 - 2.85i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.47 - 0.662i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.04 - 1.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.950 - 1.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.00 - 0.805i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.09iT - 71T^{2} \)
73 \( 1 + (9.41 - 2.52i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.03 + 6.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.99 + 5.99i)T - 83iT^{2} \)
89 \( 1 + (1.77 - 1.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.63 + 6.63i)T - 97iT^{2} \)
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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

−12.88120968307813886492066852839, −11.58360967138037674862505275578, −11.27798601235586585813193478671, −10.17034122613634442683696646460, −8.572523167707134855414626283906, −7.40049977766099024465879988267, −6.21604621875118952188726811983, −4.57589192761815399073806721069, −3.51224841612126703131525468894, −1.38752120652624158726527407363, 3.34609269282438897258670000837, 4.81476647729777080752246346009, 5.28742000415818397164891160602, 7.17972057534044035819722475560, 8.080605094972008249151877324500, 8.958815851149047985943982588450, 10.64056719566150838658311488263, 11.54552818440653908603588432258, 12.67286643068164769210244540967, 13.39051513254295118291872910049

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