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Properties

Label 2-726-33.17-c1-0-18
Degree $2$
Conductor $726$
Sign $0.929 + 0.369i$
Analytic cond. $5.79713$
Root an. cond. $2.40772$
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  + (0.309 + 0.951i)2-s + (−0.751 − 1.56i)3-s + (−0.809 + 0.587i)4-s + (0.716 + 0.232i)5-s + (1.25 − 1.19i)6-s + (−0.273 − 0.376i)7-s + (−0.809 − 0.587i)8-s + (−1.86 + 2.34i)9-s + 0.753i·10-s + (1.52 + 0.820i)12-s + (5.05 − 1.64i)13-s + (0.273 − 0.376i)14-s + (−0.175 − 1.29i)15-s + (0.309 − 0.951i)16-s + (1.14 − 3.51i)17-s + (−2.80 − 1.05i)18-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.434 − 0.900i)3-s + (−0.404 + 0.293i)4-s + (0.320 + 0.104i)5-s + (0.511 − 0.488i)6-s + (−0.103 − 0.142i)7-s + (−0.286 − 0.207i)8-s + (−0.623 + 0.782i)9-s + 0.238i·10-s + (0.440 + 0.236i)12-s + (1.40 − 0.455i)13-s + (0.0731 − 0.100i)14-s + (−0.0452 − 0.333i)15-s + (0.0772 − 0.237i)16-s + (0.277 − 0.853i)17-s + (−0.662 − 0.248i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $0.929 + 0.369i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ 0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43956 - 0.275540i\)
\(L(\frac12)\) \(\approx\) \(1.43956 - 0.275540i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.751 + 1.56i)T \)
11 \( 1 \)
good5 \( 1 + (-0.716 - 0.232i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.273 + 0.376i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-5.05 + 1.64i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.14 + 3.51i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.25 + 1.72i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 6.24iT - 23T^{2} \)
29 \( 1 + (-8.11 + 5.89i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.13 + 3.48i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.66 + 4.11i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.0353 - 0.0257i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.49iT - 43T^{2} \)
47 \( 1 + (1.87 - 2.58i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.86 + 1.58i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.83 - 9.40i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.75 + 1.21i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 6.70T + 67T^{2} \)
71 \( 1 + (-0.854 - 0.277i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.52 - 6.23i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.78 + 1.23i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.34 - 10.3i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 6.48iT - 89T^{2} \)
97 \( 1 + (0.817 + 2.51i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.36189653363193727736896899521, −9.369860317100507332258775866361, −8.307505485740193965768729079734, −7.64386409692266560893081873064, −6.75421576179532159633660936346, −5.94483142075574697055943458240, −5.34925962746124921811955598251, −3.94840697321950893892982248319, −2.58649163417704968815280346072, −0.890166059390733697026589040515, 1.33169728091595032092236977261, 3.02470156702244872795848902261, 3.94816596040186669011376492426, 4.84329622302627437528387714543, 5.90579599624126337243668033535, 6.48657293014645432880540264081, 8.300545002689918095202449540553, 8.912818411975462608334408963579, 9.824791286182510406162924951891, 10.53443909003161937103869089645

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