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Properties

Label 2-726-33.2-c1-0-15
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: $\chi_{726} (233, \cdot )$
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

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

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