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Properties

Label 2-7e3-49.37-c1-0-16
Degree $2$
Conductor $343$
Sign $-0.159 + 0.987i$
Analytic cond. $2.73886$
Root an. cond. $1.65495$
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  + (1.63 − 1.11i)2-s + (1.94 − 1.80i)3-s + (0.698 − 1.77i)4-s + (−1.54 − 0.476i)5-s + (1.16 − 5.11i)6-s + (0.0391 + 0.171i)8-s + (0.301 − 4.02i)9-s + (−3.05 + 0.941i)10-s + (0.330 + 4.41i)11-s + (−1.85 − 4.71i)12-s + (1.03 + 0.498i)13-s + (−3.86 + 1.85i)15-s + (3.05 + 2.83i)16-s + (−6.42 − 0.968i)17-s + (−3.99 − 6.91i)18-s + (0.938 − 1.62i)19-s + ⋯
L(s)  = 1  + (1.15 − 0.787i)2-s + (1.12 − 1.04i)3-s + (0.349 − 0.889i)4-s + (−0.690 − 0.212i)5-s + (0.476 − 2.08i)6-s + (0.0138 + 0.0605i)8-s + (0.100 − 1.34i)9-s + (−0.965 + 0.297i)10-s + (0.0997 + 1.33i)11-s + (−0.534 − 1.36i)12-s + (0.287 + 0.138i)13-s + (−0.996 + 0.480i)15-s + (0.764 + 0.709i)16-s + (−1.55 − 0.234i)17-s + (−0.940 − 1.62i)18-s + (0.215 − 0.372i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $-0.159 + 0.987i$
Analytic conductor: \(2.73886\)
Root analytic conductor: \(1.65495\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{343} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 343,\ (\ :1/2),\ -0.159 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89162 - 2.22200i\)
\(L(\frac12)\) \(\approx\) \(1.89162 - 2.22200i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-1.63 + 1.11i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (-1.94 + 1.80i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (1.54 + 0.476i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (-0.330 - 4.41i)T + (-10.8 + 1.63i)T^{2} \)
13 \( 1 + (-1.03 - 0.498i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (6.42 + 0.968i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-0.938 + 1.62i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.292 + 0.0440i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (-1.70 + 2.13i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-2.96 - 5.14i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.31 + 8.45i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-0.655 - 2.87i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (0.200 - 0.878i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (1.40 - 0.958i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (0.791 - 2.01i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-2.62 + 0.809i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (1.18 + 3.01i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-1.66 - 2.88i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.37 + 9.25i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (4.72 + 3.22i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (-5.25 + 9.10i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.48 - 3.12i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (1.01 - 13.5i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 3.26T + 97T^{2} \)
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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

−11.75384896338955808647433454341, −10.69478262337019024420762947316, −9.270950853367805105367741205963, −8.366528294229920408096620451262, −7.45557371086522447144556024190, −6.54065653569245561874305857031, −4.79911694166683231940665681169, −3.98138202849713013725699092870, −2.72653855103317705552831995285, −1.81874717644909164055781540798, 3.00633484117351128480108816591, 3.79355832166775182186994456062, 4.51324845583860934228913027004, 5.77268177188957918451775097833, 6.86117697442867118637854092157, 8.106025586578720749202454323060, 8.694880840858424278638262366952, 9.855344054140434811875171287306, 10.89392084088416722194160044594, 11.81570746058608807728018891967

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