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Properties

Label 2-740-148.31-c1-0-53
Degree $2$
Conductor $740$
Sign $0.759 + 0.650i$
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  + (−1.38 − 0.282i)2-s + 3.03·3-s + (1.84 + 0.782i)4-s + (−0.707 + 0.707i)5-s + (−4.20 − 0.855i)6-s − 4.17i·7-s + (−2.33 − 1.60i)8-s + 6.18·9-s + (1.17 − 0.780i)10-s + 2.56·11-s + (5.57 + 2.37i)12-s + (1.37 − 1.37i)13-s + (−1.17 + 5.78i)14-s + (−2.14 + 2.14i)15-s + (2.77 + 2.87i)16-s + (−2.87 + 2.87i)17-s + ⋯
L(s)  = 1  + (−0.979 − 0.199i)2-s + 1.75·3-s + (0.920 + 0.391i)4-s + (−0.316 + 0.316i)5-s + (−1.71 − 0.349i)6-s − 1.57i·7-s + (−0.823 − 0.566i)8-s + 2.06·9-s + (0.372 − 0.246i)10-s + 0.773·11-s + (1.61 + 0.684i)12-s + (0.380 − 0.380i)13-s + (−0.314 + 1.54i)14-s + (−0.553 + 0.553i)15-s + (0.694 + 0.719i)16-s + (−0.696 + 0.696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.759 + 0.650i$
Analytic conductor: \(5.90892\)
Root analytic conductor: \(2.43082\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :1/2),\ 0.759 + 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69486 - 0.626697i\)
\(L(\frac12)\) \(\approx\) \(1.69486 - 0.626697i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.282i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (2.30 - 5.62i)T \)
good3 \( 1 - 3.03T + 3T^{2} \)
7 \( 1 + 4.17iT - 7T^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
13 \( 1 + (-1.37 + 1.37i)T - 13iT^{2} \)
17 \( 1 + (2.87 - 2.87i)T - 17iT^{2} \)
19 \( 1 + (-3.33 - 3.33i)T + 19iT^{2} \)
23 \( 1 + (5.11 + 5.11i)T + 23iT^{2} \)
29 \( 1 + (0.274 + 0.274i)T + 29iT^{2} \)
31 \( 1 + (-4.00 + 4.00i)T - 31iT^{2} \)
41 \( 1 + 8.34iT - 41T^{2} \)
43 \( 1 + (3.95 + 3.95i)T + 43iT^{2} \)
47 \( 1 - 7.51iT - 47T^{2} \)
53 \( 1 - 0.614T + 53T^{2} \)
59 \( 1 + (-6.70 - 6.70i)T + 59iT^{2} \)
61 \( 1 + (-2.54 - 2.54i)T + 61iT^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 - 7.59iT - 71T^{2} \)
73 \( 1 + 9.40iT - 73T^{2} \)
79 \( 1 + (-1.43 - 1.43i)T + 79iT^{2} \)
83 \( 1 - 16.9iT - 83T^{2} \)
89 \( 1 + (-3.33 - 3.33i)T + 89iT^{2} \)
97 \( 1 + (7.76 - 7.76i)T - 97iT^{2} \)
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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

−10.17833260813067798246171345595, −9.415031505440354100678917215890, −8.337556333892629582053360280256, −8.049758880290899325517829656414, −7.14802035141923726949154217405, −6.47504308662864363048601956096, −3.91164736174240607718132075618, −3.79766255434391924569979229888, −2.44277704936227454780302474316, −1.19504665131965136909309887764, 1.66909099143354333745508853427, 2.57524153188513795745233993918, 3.56463755166363139431104313777, 5.09147650980438117904171075272, 6.42672517922641462257793009341, 7.32214263062884646771848174190, 8.308570842023909871598199871164, 8.712130307047920811546024215668, 9.385916304771610990615508316631, 9.780058928539246721765160079024

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