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Properties

Label 2-731-731.208-c1-0-61
Degree $2$
Conductor $731$
Sign $-0.939 - 0.343i$
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  − 1.83i·2-s + (1.97 + 0.528i)3-s − 1.37·4-s + (−3.10 − 0.832i)5-s + (0.970 − 3.62i)6-s + (−0.484 + 0.129i)7-s − 1.14i·8-s + (1.01 + 0.583i)9-s + (−1.53 + 5.71i)10-s + (−2.25 − 2.25i)11-s + (−2.71 − 0.726i)12-s + (−1.24 + 2.15i)13-s + (0.238 + 0.890i)14-s + (−5.68 − 3.28i)15-s − 4.85·16-s + (−4.11 − 0.286i)17-s + ⋯
L(s)  = 1  − 1.29i·2-s + (1.13 + 0.305i)3-s − 0.687·4-s + (−1.39 − 0.372i)5-s + (0.396 − 1.47i)6-s + (−0.183 + 0.0491i)7-s − 0.405i·8-s + (0.336 + 0.194i)9-s + (−0.483 + 1.80i)10-s + (−0.680 − 0.680i)11-s + (−0.782 − 0.209i)12-s + (−0.344 + 0.597i)13-s + (0.0638 + 0.238i)14-s + (−1.46 − 0.848i)15-s − 1.21·16-s + (−0.997 − 0.0694i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.939 - 0.343i$
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: \((2,\ 731,\ (\ :1/2),\ -0.939 - 0.343i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.181541 + 1.02624i\)
\(L(\frac12)\) \(\approx\) \(0.181541 + 1.02624i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (4.11 + 0.286i)T \)
43 \( 1 + (2.60 + 6.01i)T \)
good2 \( 1 + 1.83iT - 2T^{2} \)
3 \( 1 + (-1.97 - 0.528i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (3.10 + 0.832i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.484 - 0.129i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (2.25 + 2.25i)T + 11iT^{2} \)
13 \( 1 + (1.24 - 2.15i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (-1.82 + 1.05i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.85 + 6.92i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.390 + 1.45i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (0.859 + 0.230i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-9.08 - 2.43i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.72 - 4.72i)T + 41iT^{2} \)
47 \( 1 - 2.32T + 47T^{2} \)
53 \( 1 + (-0.677 + 0.391i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.90iT - 59T^{2} \)
61 \( 1 + (-1.47 + 0.394i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.89 - 5.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.417 + 1.55i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.44 + 5.38i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-12.7 + 3.41i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (10.6 - 6.14i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.94 - 5.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.80 - 5.80i)T - 97iT^{2} \)
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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

−9.890189906467104217790802202155, −9.113194824851588112384384564516, −8.457632755425101386172838372896, −7.68983972489329453852008253429, −6.54764527991936509722592593961, −4.67135968608514619019309749148, −4.02553694426402390246346324125, −3.08926546140884969579166428687, −2.39254991633665438043705661100, −0.43884073692128527581862501583, 2.39337803181498257126399371903, 3.40416589569150613074168309596, 4.56310535097000399376800073317, 5.66302519803336889508793901365, 6.98316607276435934052861651568, 7.57387573004897177707583797220, 7.85908388981163746634632831023, 8.728673804831302932510040172974, 9.620591171446667847989571253834, 10.95188773334325888485790923545

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