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Properties

Label 2-740-185.17-c1-0-12
Degree $2$
Conductor $740$
Sign $0.994 + 0.103i$
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  + (0.0586 − 0.0410i)3-s + (1.91 + 1.15i)5-s + (−0.252 − 2.89i)7-s + (−1.02 + 2.81i)9-s + (3.18 − 1.83i)11-s + (0.615 − 0.224i)13-s + (0.159 − 0.0108i)15-s + (1.54 − 4.25i)17-s + (1.52 + 2.17i)19-s + (−0.133 − 0.159i)21-s + (0.987 − 1.71i)23-s + (2.33 + 4.42i)25-s + (0.111 + 0.414i)27-s + (−1.86 + 6.94i)29-s + (3.49 − 3.49i)31-s + ⋯
L(s)  = 1  + (0.0338 − 0.0237i)3-s + (0.856 + 0.516i)5-s + (−0.0956 − 1.09i)7-s + (−0.341 + 0.938i)9-s + (0.958 − 0.553i)11-s + (0.170 − 0.0621i)13-s + (0.0412 − 0.00281i)15-s + (0.375 − 1.03i)17-s + (0.349 + 0.498i)19-s + (−0.0291 − 0.0347i)21-s + (0.206 − 0.356i)23-s + (0.466 + 0.884i)25-s + (0.0213 + 0.0797i)27-s + (−0.345 + 1.29i)29-s + (0.627 − 0.627i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.994 + 0.103i$
Analytic conductor: \(5.90892\)
Root analytic conductor: \(2.43082\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :1/2),\ 0.994 + 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85752 - 0.0963674i\)
\(L(\frac12)\) \(\approx\) \(1.85752 - 0.0963674i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.91 - 1.15i)T \)
37 \( 1 + (-5.78 - 1.87i)T \)
good3 \( 1 + (-0.0586 + 0.0410i)T + (1.02 - 2.81i)T^{2} \)
7 \( 1 + (0.252 + 2.89i)T + (-6.89 + 1.21i)T^{2} \)
11 \( 1 + (-3.18 + 1.83i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.615 + 0.224i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.54 + 4.25i)T + (-13.0 - 10.9i)T^{2} \)
19 \( 1 + (-1.52 - 2.17i)T + (-6.49 + 17.8i)T^{2} \)
23 \( 1 + (-0.987 + 1.71i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.86 - 6.94i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-3.49 + 3.49i)T - 31iT^{2} \)
41 \( 1 + (2.44 + 6.72i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + 0.409T + 43T^{2} \)
47 \( 1 + (-0.396 + 0.106i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.141 - 1.62i)T + (-52.1 - 9.20i)T^{2} \)
59 \( 1 + (-0.0782 + 0.893i)T + (-58.1 - 10.2i)T^{2} \)
61 \( 1 + (2.56 - 5.50i)T + (-39.2 - 46.7i)T^{2} \)
67 \( 1 + (-3.82 + 0.334i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (1.24 - 7.06i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (2.25 + 2.25i)T + 73iT^{2} \)
79 \( 1 + (-4.33 + 0.379i)T + (77.7 - 13.7i)T^{2} \)
83 \( 1 + (0.740 - 0.345i)T + (53.3 - 63.5i)T^{2} \)
89 \( 1 + (-3.98 - 0.348i)T + (87.6 + 15.4i)T^{2} \)
97 \( 1 + (1.88 + 1.08i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.40354274811515365942379791513, −9.626476173336780088020068775115, −8.729536782139677886658175450309, −7.60125742492064664818970765012, −6.91244833459039645475768920143, −5.96005104729814897938462372827, −5.02489858321880112391769637238, −3.75543978209197873190389766968, −2.69777288676869212432863564483, −1.22763690558821942714521705116, 1.34467028425489527819176977891, 2.60353228689493577073462647150, 3.90042047052548901103997904560, 5.10327262257386755687920374911, 6.10009672218633721785252875340, 6.48545168239964676986018468472, 8.006493669198425720549252147770, 8.974369182517104578048630009890, 9.357282145192622434507644834756, 10.09657766411456913779885881875

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