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Properties

Label 2-731-731.208-c1-0-3
Degree $2$
Conductor $731$
Sign $0.904 - 0.425i$
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  − 1.79i·2-s + (−1.99 − 0.534i)3-s − 1.22·4-s + (0.473 + 0.126i)5-s + (−0.960 + 3.58i)6-s + (−3.30 + 0.885i)7-s − 1.38i·8-s + (1.09 + 0.631i)9-s + (0.228 − 0.851i)10-s + (−1.62 − 1.62i)11-s + (2.45 + 0.657i)12-s + (−0.566 + 0.981i)13-s + (1.59 + 5.93i)14-s + (−0.877 − 0.506i)15-s − 4.94·16-s + (3.68 + 1.84i)17-s + ⋯
L(s)  = 1  − 1.27i·2-s + (−1.15 − 0.308i)3-s − 0.614·4-s + (0.211 + 0.0567i)5-s + (−0.392 + 1.46i)6-s + (−1.24 + 0.334i)7-s − 0.489i·8-s + (0.364 + 0.210i)9-s + (0.0721 − 0.269i)10-s + (−0.489 − 0.489i)11-s + (0.708 + 0.189i)12-s + (−0.157 + 0.272i)13-s + (0.425 + 1.58i)14-s + (−0.226 − 0.130i)15-s − 1.23·16-s + (0.894 + 0.447i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.904 - 0.425i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.904 - 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.280244 + 0.0626722i\)
\(L(\frac12)\) \(\approx\) \(0.280244 + 0.0626722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.68 - 1.84i)T \)
43 \( 1 + (6.55 - 0.144i)T \)
good2 \( 1 + 1.79iT - 2T^{2} \)
3 \( 1 + (1.99 + 0.534i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.473 - 0.126i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (3.30 - 0.885i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.62 + 1.62i)T + 11iT^{2} \)
13 \( 1 + (0.566 - 0.981i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (-0.798 + 0.461i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.50 - 5.61i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.39 - 5.22i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-4.40 - 1.18i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-6.30 - 1.68i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.10 - 4.10i)T + 41iT^{2} \)
47 \( 1 + 7.21T + 47T^{2} \)
53 \( 1 + (3.85 - 2.22i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.52iT - 59T^{2} \)
61 \( 1 + (0.621 - 0.166i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.90 - 5.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.62 + 13.5i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.02 + 3.81i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.63 + 1.24i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (8.63 - 4.98i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.61 - 6.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.50 - 6.50i)T - 97iT^{2} \)
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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

−10.51587363808041975809464047681, −9.918143976050399453766463422824, −9.242646675575413160608323648195, −7.84367235315046869474149400853, −6.54604751817991765188325470861, −6.11431754720244001840505185705, −5.06558103565689094751940199037, −3.57497823517190658031055845404, −2.79450357880642310530436866800, −1.27088922377658143375686943863, 0.18033511471269033823568392688, 2.71492772582034706494368619121, 4.33606043888028420257381882567, 5.29665002176931255481688342728, 5.99178076690959908264817233531, 6.57765357928189794321832649409, 7.49376851184473250832409753956, 8.317065007416163947332264542856, 9.782275625081309075824680567926, 10.00783230404205094806336133438

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