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Properties

Label 2-731-17.4-c1-0-28
Degree $2$
Conductor $731$
Sign $-0.263 - 0.964i$
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  + 0.831i·2-s + (−0.711 + 0.711i)3-s + 1.30·4-s + (0.619 − 0.619i)5-s + (−0.591 − 0.591i)6-s + (2.80 + 2.80i)7-s + 2.75i·8-s + 1.98i·9-s + (0.514 + 0.514i)10-s + (−1.65 − 1.65i)11-s + (−0.932 + 0.932i)12-s + 0.606·13-s + (−2.33 + 2.33i)14-s + 0.881i·15-s + 0.333·16-s + (1.58 − 3.80i)17-s + ⋯
L(s)  = 1  + 0.587i·2-s + (−0.410 + 0.410i)3-s + 0.654·4-s + (0.276 − 0.276i)5-s + (−0.241 − 0.241i)6-s + (1.06 + 1.06i)7-s + 0.972i·8-s + 0.662i·9-s + (0.162 + 0.162i)10-s + (−0.497 − 0.497i)11-s + (−0.269 + 0.269i)12-s + 0.168·13-s + (−0.623 + 0.623i)14-s + 0.227i·15-s + 0.0833·16-s + (0.385 − 0.922i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.263 - 0.964i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.263 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10852 + 1.45260i\)
\(L(\frac12)\) \(\approx\) \(1.10852 + 1.45260i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.58 + 3.80i)T \)
43 \( 1 - iT \)
good2 \( 1 - 0.831iT - 2T^{2} \)
3 \( 1 + (0.711 - 0.711i)T - 3iT^{2} \)
5 \( 1 + (-0.619 + 0.619i)T - 5iT^{2} \)
7 \( 1 + (-2.80 - 2.80i)T + 7iT^{2} \)
11 \( 1 + (1.65 + 1.65i)T + 11iT^{2} \)
13 \( 1 - 0.606T + 13T^{2} \)
19 \( 1 + 0.918iT - 19T^{2} \)
23 \( 1 + (0.0627 + 0.0627i)T + 23iT^{2} \)
29 \( 1 + (3.46 - 3.46i)T - 29iT^{2} \)
31 \( 1 + (-3.62 + 3.62i)T - 31iT^{2} \)
37 \( 1 + (-1.28 + 1.28i)T - 37iT^{2} \)
41 \( 1 + (-0.348 - 0.348i)T + 41iT^{2} \)
47 \( 1 + 2.60T + 47T^{2} \)
53 \( 1 + 4.50iT - 53T^{2} \)
59 \( 1 - 8.50iT - 59T^{2} \)
61 \( 1 + (5.64 + 5.64i)T + 61iT^{2} \)
67 \( 1 - 5.37T + 67T^{2} \)
71 \( 1 + (-1.41 + 1.41i)T - 71iT^{2} \)
73 \( 1 + (8.64 - 8.64i)T - 73iT^{2} \)
79 \( 1 + (-7.96 - 7.96i)T + 79iT^{2} \)
83 \( 1 + 16.4iT - 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + (-8.56 + 8.56i)T - 97iT^{2} \)
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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

−10.92541547634517840544982456425, −9.773987528413955168743047808594, −8.714474891143270335734520755836, −8.018793567537144594447153627252, −7.25357866182296563009454862802, −5.87786888081815626233517459788, −5.42065837639561644557770620512, −4.76142094182725757340766919718, −2.86052895778777220081621236895, −1.81924427914506187204455806417, 1.06306616958310406736297611578, 2.04508359219492080428130024669, 3.47169246631881924875360924213, 4.47520008134136463031777819308, 5.86239019774212405354689910166, 6.64730129870016047611963301719, 7.45902538341145203640658071236, 8.188966616793287306858171289335, 9.678861890219340324857752357326, 10.42601559736614278834415863747

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