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Properties

Label 2-7e3-49.15-c1-0-9
Degree $2$
Conductor $343$
Sign $0.717 - 0.696i$
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  + (−0.270 + 1.18i)2-s + (0.324 − 0.407i)3-s + (0.468 + 0.225i)4-s + (1.61 − 2.02i)5-s + (0.394 + 0.495i)6-s + (−1.91 + 2.39i)8-s + (0.607 + 2.66i)9-s + (1.96 + 2.46i)10-s + (0.855 − 3.74i)11-s + (0.243 − 0.117i)12-s + (−0.203 + 0.891i)13-s + (−0.300 − 1.31i)15-s + (−1.67 − 2.10i)16-s + (4.44 − 2.14i)17-s − 3.32·18-s + 2.49·19-s + ⋯
L(s)  = 1  + (−0.191 + 0.838i)2-s + (0.187 − 0.234i)3-s + (0.234 + 0.112i)4-s + (0.723 − 0.907i)5-s + (0.161 + 0.202i)6-s + (−0.675 + 0.847i)8-s + (0.202 + 0.886i)9-s + (0.622 + 0.780i)10-s + (0.257 − 1.12i)11-s + (0.0703 − 0.0338i)12-s + (−0.0564 + 0.247i)13-s + (−0.0776 − 0.340i)15-s + (−0.419 − 0.525i)16-s + (1.07 − 0.519i)17-s − 0.782·18-s + 0.571·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $0.717 - 0.696i$
Analytic conductor: \(2.73886\)
Root analytic conductor: \(1.65495\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{343} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 343,\ (\ :1/2),\ 0.717 - 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49890 + 0.607908i\)
\(L(\frac12)\) \(\approx\) \(1.49890 + 0.607908i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.270 - 1.18i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (-0.324 + 0.407i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (-1.61 + 2.02i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (-0.855 + 3.74i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.203 - 0.891i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-4.44 + 2.14i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 - 2.49T + 19T^{2} \)
23 \( 1 + (-3.55 - 1.71i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (4.93 - 2.37i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + 9.86T + 31T^{2} \)
37 \( 1 + (1.97 - 0.951i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (1.31 - 1.65i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (2.67 + 3.35i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-1.27 + 5.57i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-6.92 - 3.33i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (6.04 + 7.58i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (13.0 - 6.27i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 - 6.76T + 67T^{2} \)
71 \( 1 + (-5.32 - 2.56i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (2.52 + 11.0i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + 5.91T + 79T^{2} \)
83 \( 1 + (0.604 + 2.64i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (0.216 + 0.948i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + 17.8T + 97T^{2} \)
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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

−11.64572888957801130947613347007, −10.75245977522229393522415053485, −9.363479432730309037970316259967, −8.764273860064945644250861236423, −7.76556885593138751225635881454, −7.02849322712239519072803952484, −5.62264966620245127949622787820, −5.26020306718786347541617219295, −3.23449185660903459556013797060, −1.64569702890858265936207606665, 1.59595743478513907971837890987, 2.86358413424167318217733185374, 3.83001827464800800443462665653, 5.62793436053950779166224705083, 6.63745135316857247082553973979, 7.41424239552360641092711852030, 9.185126292623784039441258695677, 9.766079390701214135523318693141, 10.38499105770454175584723646055, 11.22905197384557801393271382921

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