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Properties

Label 2-731-17.2-c1-0-3
Degree $2$
Conductor $731$
Sign $0.780 - 0.625i$
Analytic cond. $5.83706$
Root an. cond. $2.41600$
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  + (−0.286 − 0.286i)2-s + (−2.99 + 1.24i)3-s − 1.83i·4-s + (−0.918 − 2.21i)5-s + (1.21 + 0.503i)6-s + (−1.74 + 4.20i)7-s + (−1.09 + 1.09i)8-s + (5.31 − 5.31i)9-s + (−0.372 + 0.898i)10-s + (−5.16 − 2.14i)11-s + (2.27 + 5.49i)12-s − 4.32i·13-s + (1.70 − 0.706i)14-s + (5.50 + 5.50i)15-s − 3.04·16-s + (−3.09 + 2.72i)17-s + ⋯
L(s)  = 1  + (−0.202 − 0.202i)2-s + (−1.72 + 0.716i)3-s − 0.917i·4-s + (−0.410 − 0.991i)5-s + (0.495 + 0.205i)6-s + (−0.658 + 1.58i)7-s + (−0.388 + 0.388i)8-s + (1.77 − 1.77i)9-s + (−0.117 + 0.284i)10-s + (−1.55 − 0.645i)11-s + (0.657 + 1.58i)12-s − 1.20i·13-s + (0.455 − 0.188i)14-s + (1.42 + 1.42i)15-s − 0.760·16-s + (−0.750 + 0.661i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.780 - 0.625i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.780 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.248344 + 0.0872529i\)
\(L(\frac12)\) \(\approx\) \(0.248344 + 0.0872529i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.09 - 2.72i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (0.286 + 0.286i)T + 2iT^{2} \)
3 \( 1 + (2.99 - 1.24i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.918 + 2.21i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.74 - 4.20i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (5.16 + 2.14i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 4.32iT - 13T^{2} \)
19 \( 1 + (1.17 + 1.17i)T + 19iT^{2} \)
23 \( 1 + (0.709 + 0.294i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-2.37 - 5.72i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-4.72 + 1.95i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-8.39 + 3.47i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.29 - 3.11i)T + (-28.9 - 28.9i)T^{2} \)
47 \( 1 - 8.64iT - 47T^{2} \)
53 \( 1 + (-3.96 - 3.96i)T + 53iT^{2} \)
59 \( 1 + (-2.40 + 2.40i)T - 59iT^{2} \)
61 \( 1 + (1.81 - 4.37i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 9.52T + 67T^{2} \)
71 \( 1 + (0.661 - 0.273i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (4.23 + 10.2i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-0.123 - 0.0511i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (2.68 + 2.68i)T + 83iT^{2} \)
89 \( 1 - 8.46iT - 89T^{2} \)
97 \( 1 + (-4.92 - 11.8i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.61073100526343775483553323009, −9.839515435493022146745820172553, −8.998533679176746936097784368029, −8.196607939711207996456109938801, −6.35773628258796345224476256253, −5.80194395309494966217171810986, −5.25296187092696099002119889324, −4.58037227177477030907935525536, −2.74870482282499615050903041596, −0.71003557066641268994540671962, 0.29516291791642415155669154661, 2.47558546467780674710673747014, 4.02051516058459382261756079098, 4.77423482338729836432845023779, 6.36523841778515809703006012944, 6.94834626575620745694254705710, 7.25793822345269894869922048052, 8.045149261162250397837369603107, 9.918670111590065617259167027192, 10.39474136831205732947890035298

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