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Properties

Label 2-740-740.687-c1-0-99
Degree $2$
Conductor $740$
Sign $-0.257 + 0.966i$
Analytic cond. $5.90892$
Root an. cond. $2.43082$
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  + (1.28 − 0.597i)2-s + (1.28 − 1.53i)4-s + (0.637 − 2.14i)5-s + (0.732 − 2.73i)8-s + (−1.92 − 2.29i)9-s + (−0.463 − 3.12i)10-s + (0.472 + 5.39i)13-s + (−0.694 − 3.93i)16-s + (0.542 − 6.20i)17-s + (−3.84 − 1.79i)18-s + (−2.46 − 3.73i)20-s + (−4.18 − 2.73i)25-s + (3.83 + 6.63i)26-s + (5.11 + 8.86i)29-s + (−3.24 − 4.63i)32-s + ⋯
L(s)  = 1  + (0.906 − 0.422i)2-s + (0.642 − 0.766i)4-s + (0.285 − 0.958i)5-s + (0.258 − 0.965i)8-s + (−0.642 − 0.766i)9-s + (−0.146 − 0.989i)10-s + (0.130 + 1.49i)13-s + (−0.173 − 0.984i)16-s + (0.131 − 1.50i)17-s + (−0.906 − 0.422i)18-s + (−0.550 − 0.834i)20-s + (−0.837 − 0.546i)25-s + (0.751 + 1.30i)26-s + (0.949 + 1.64i)29-s + (−0.573 − 0.819i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $-0.257 + 0.966i$
Analytic conductor: \(5.90892\)
Root analytic conductor: \(2.43082\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :1/2),\ -0.257 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56695 - 2.03880i\)
\(L(\frac12)\) \(\approx\) \(1.56695 - 2.03880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 + 0.597i)T \)
5 \( 1 + (-0.637 + 2.14i)T \)
37 \( 1 + (5.23 + 3.09i)T \)
good3 \( 1 + (1.92 + 2.29i)T^{2} \)
7 \( 1 + (-2.39 + 6.57i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.472 - 5.39i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (-0.542 + 6.20i)T + (-16.7 - 2.95i)T^{2} \)
19 \( 1 + (-14.5 + 12.2i)T^{2} \)
23 \( 1 + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-5.11 - 8.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
41 \( 1 + (-6.47 - 5.43i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-11.9 - 8.35i)T + (18.1 + 49.8i)T^{2} \)
59 \( 1 + (55.4 - 20.1i)T^{2} \)
61 \( 1 + (-9.88 + 11.7i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-22.9 + 62.9i)T^{2} \)
71 \( 1 + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (7.02 - 7.02i)T - 73iT^{2} \)
79 \( 1 + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (81.7 + 14.4i)T^{2} \)
89 \( 1 + (3.03 + 17.2i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-4.82 - 17.9i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.13911113464811028309525378732, −9.232083967158910775087867100190, −8.789177496422459304360083942856, −7.19947579629342894442730845259, −6.42619890910964948287024023864, −5.40793474718654205120226728573, −4.68777245983416043791190422193, −3.65915162725328727372815091538, −2.41493496929634844991592573902, −1.02843916412949631929001742572, 2.26255147572216283367993060117, 3.11267546231470483342148981600, 4.17161433105847601864380270572, 5.60974299050905943408367020235, 5.89236923530136892359230930813, 7.01947063096253432288839798395, 7.958151337972861204255645584294, 8.467372688722847251224578091841, 10.20867246886907358919670694986, 10.60971762600409558343364987111

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