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Properties

Label 2-731-43.35-c1-0-27
Degree $2$
Conductor $731$
Sign $0.893 - 0.449i$
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  + (−0.257 + 1.12i)2-s + (−0.136 − 0.597i)3-s + (0.593 + 0.285i)4-s + (1.65 − 2.07i)5-s + 0.709·6-s − 2.51·7-s + (−1.91 + 2.40i)8-s + (2.36 − 1.13i)9-s + (1.91 + 2.40i)10-s + (1.70 − 0.823i)11-s + (0.0897 − 0.393i)12-s + (−2.39 + 2.99i)13-s + (0.648 − 2.84i)14-s + (−1.46 − 0.706i)15-s + (−1.40 − 1.75i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (−0.182 + 0.798i)2-s + (−0.0786 − 0.344i)3-s + (0.296 + 0.142i)4-s + (0.740 − 0.928i)5-s + 0.289·6-s − 0.951·7-s + (−0.678 + 0.851i)8-s + (0.788 − 0.379i)9-s + (0.606 + 0.760i)10-s + (0.515 − 0.248i)11-s + (0.0259 − 0.113i)12-s + (−0.663 + 0.831i)13-s + (0.173 − 0.759i)14-s + (−0.378 − 0.182i)15-s + (−0.350 − 0.439i)16-s + (0.151 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.893 - 0.449i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.893 - 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66667 + 0.395808i\)
\(L(\frac12)\) \(\approx\) \(1.66667 + 0.395808i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (-5.75 + 3.14i)T \)
good2 \( 1 + (0.257 - 1.12i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (0.136 + 0.597i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (-1.65 + 2.07i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + 2.51T + 7T^{2} \)
11 \( 1 + (-1.70 + 0.823i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (2.39 - 2.99i)T + (-2.89 - 12.6i)T^{2} \)
19 \( 1 + (-6.65 - 3.20i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-7.09 + 3.41i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (-0.0737 + 0.323i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.66 + 7.30i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 - 6.55T + 37T^{2} \)
41 \( 1 + (-1.71 + 7.50i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (4.99 + 2.40i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-3.03 - 3.80i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-5.42 - 6.80i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-1.55 - 6.81i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (7.14 + 3.44i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (9.47 + 4.56i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (9.85 - 12.3i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + (-0.742 - 3.25i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-2.76 - 12.1i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (17.1 - 8.26i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.09023615221914135367084973355, −9.351933555661888463722275838895, −8.889954448257265929805668525371, −7.56956732798056478987942398799, −6.97690372038625416544811634355, −6.14728076688153820128023482421, −5.44154478838684566355448402661, −4.10631983961080134854433624093, −2.66101233133495939238571255167, −1.19937302544933644113264134559, 1.29231463501860415119394197249, 2.86243643662521707093127386445, 3.16466438977031966098508297283, 4.84510828386663518515297039526, 5.96349935486778603503025393576, 6.91283212048814621517199402451, 7.36534494395536403640769902295, 9.269428889933286379487499676194, 9.793807314871955252622270188230, 10.15610869581108504053642227182

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