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Properties

Label 2-7e3-49.32-c1-0-9
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  + (−0.0546 − 0.728i)2-s + (−1.09 − 0.338i)3-s + (1.44 − 0.218i)4-s + (−0.189 + 0.175i)5-s + (−0.186 + 0.818i)6-s + (−0.563 − 2.46i)8-s + (−1.38 − 0.945i)9-s + (0.138 + 0.128i)10-s + (3.10 − 2.11i)11-s + (−1.66 − 0.251i)12-s + (2.38 + 1.15i)13-s + (0.267 − 0.129i)15-s + (1.03 − 0.318i)16-s + (−1.87 − 4.78i)17-s + (−0.613 + 1.06i)18-s + (−1.91 − 3.32i)19-s + ⋯
L(s)  = 1  + (−0.0386 − 0.515i)2-s + (−0.634 − 0.195i)3-s + (0.724 − 0.109i)4-s + (−0.0847 + 0.0786i)5-s + (−0.0762 + 0.334i)6-s + (−0.199 − 0.872i)8-s + (−0.462 − 0.315i)9-s + (0.0438 + 0.0406i)10-s + (0.936 − 0.638i)11-s + (−0.481 − 0.0725i)12-s + (0.662 + 0.319i)13-s + (0.0691 − 0.0333i)15-s + (0.258 − 0.0797i)16-s + (−0.455 − 1.16i)17-s + (−0.144 + 0.250i)18-s + (−0.440 − 0.762i)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: $\chi_{343} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 343,\ (\ :1/2),\ -0.159 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.782968 - 0.919719i\)
\(L(\frac12)\) \(\approx\) \(0.782968 - 0.919719i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0546 + 0.728i)T + (-1.97 + 0.298i)T^{2} \)
3 \( 1 + (1.09 + 0.338i)T + (2.47 + 1.68i)T^{2} \)
5 \( 1 + (0.189 - 0.175i)T + (0.373 - 4.98i)T^{2} \)
11 \( 1 + (-3.10 + 2.11i)T + (4.01 - 10.2i)T^{2} \)
13 \( 1 + (-2.38 - 1.15i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (1.87 + 4.78i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (1.91 + 3.32i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.441 + 1.12i)T + (-16.8 - 15.6i)T^{2} \)
29 \( 1 + (-0.581 + 0.728i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-2.14 + 3.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.00 + 0.603i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (1.90 + 8.34i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (1.08 - 4.75i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-0.894 - 11.9i)T + (-46.4 + 7.00i)T^{2} \)
53 \( 1 + (-3.79 + 0.571i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (-3.94 - 3.65i)T + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (-13.5 - 2.03i)T + (58.2 + 17.9i)T^{2} \)
67 \( 1 + (7.19 - 12.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.94 - 6.19i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (0.630 - 8.40i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (-4.32 - 7.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.3 + 4.98i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-6.02 - 4.10i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + 8.15T + 97T^{2} \)
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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

−11.47537845873871140521814741281, −10.76585680077582040178646508601, −9.462764970508186026620853087946, −8.683534868968841372338728612225, −7.07801956306353253653786360729, −6.50419401287026802220625843877, −5.54369101495019014065220465127, −3.91076155924536580656637279602, −2.68650228767942358310187478638, −0.962815403803637787259551224716, 1.93819466747693249790903774787, 3.68620424147820616358300180710, 5.06706359851774452613842495094, 6.18658546736406971298693171578, 6.66708020390088492816206368379, 8.068766939222535661218587187007, 8.654934222236447882076863844895, 10.24300340484681120256517943813, 10.82286130739544552843352513582, 11.83032594569770447188891401684

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