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Properties

Label 2-140-140.23-c1-0-11
Degree $2$
Conductor $140$
Sign $0.976 + 0.216i$
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  + (−1.41 − 0.0671i)2-s + (2.38 + 0.638i)3-s + (1.99 + 0.189i)4-s + (−0.525 − 2.17i)5-s + (−3.32 − 1.06i)6-s + (2.10 − 1.60i)7-s + (−2.79 − 0.401i)8-s + (2.68 + 1.54i)9-s + (0.596 + 3.10i)10-s + (−4.09 + 2.36i)11-s + (4.62 + 1.72i)12-s + (0.0592 + 0.0592i)13-s + (−3.07 + 2.13i)14-s + (0.135 − 5.51i)15-s + (3.92 + 0.755i)16-s + (4.77 + 1.27i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0475i)2-s + (1.37 + 0.368i)3-s + (0.995 + 0.0948i)4-s + (−0.235 − 0.971i)5-s + (−1.35 − 0.433i)6-s + (0.794 − 0.607i)7-s + (−0.989 − 0.142i)8-s + (0.893 + 0.515i)9-s + (0.188 + 0.982i)10-s + (−1.23 + 0.713i)11-s + (1.33 + 0.497i)12-s + (0.0164 + 0.0164i)13-s + (−0.822 + 0.569i)14-s + (0.0349 − 1.42i)15-s + (0.981 + 0.188i)16-s + (1.15 + 0.310i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.976 + 0.216i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.976 + 0.216i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06628 - 0.116870i\)
\(L(\frac12)\) \(\approx\) \(1.06628 - 0.116870i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0671i)T \)
5 \( 1 + (0.525 + 2.17i)T \)
7 \( 1 + (-2.10 + 1.60i)T \)
good3 \( 1 + (-2.38 - 0.638i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (4.09 - 2.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0592 - 0.0592i)T + 13iT^{2} \)
17 \( 1 + (-4.77 - 1.27i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.31 - 2.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.292 + 1.09i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 7.27iT - 29T^{2} \)
31 \( 1 + (4.01 - 2.31i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.596 + 2.22i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.71T + 41T^{2} \)
43 \( 1 + (1.57 - 1.57i)T - 43iT^{2} \)
47 \( 1 + (-2.67 + 0.716i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.44 + 9.12i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.67 + 2.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.978 - 1.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.131 - 0.491i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + (2.48 - 9.26i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.05 + 5.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.62 - 5.62i)T - 83iT^{2} \)
89 \( 1 + (-14.4 - 8.34i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.81 + 5.81i)T - 97iT^{2} \)
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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

−13.06814069768588685303855904584, −12.12955570824682063760466183211, −10.63558785047521607472508865827, −9.890432422418368087715307805403, −8.775069793294707597965629356560, −8.068448458806000849595739227193, −7.44418413118122835092665338768, −5.14648812933714876373486855903, −3.56095816813702230460358758299, −1.82301507870924112188268592943, 2.27118370660097191600630362364, 3.15233324436793362119113049122, 5.78563451884278796160199225978, 7.42402296567634703799334752304, 7.935198710357148961459555872974, 8.761064029626651301972973731934, 9.963113549771649040774775533640, 10.98880497842392563776085791071, 11.93314895236731773537421932166, 13.42784104009339340559224792769

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