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Properties

Label 2-7e3-49.29-c1-0-13
Degree $2$
Conductor $343$
Sign $-0.927 + 0.372i$
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  + (−0.794 − 0.382i)2-s + (0.0628 + 0.275i)3-s + (−0.762 − 0.956i)4-s + (−0.452 − 1.98i)5-s + (0.0553 − 0.242i)6-s + (0.632 + 2.76i)8-s + (2.63 − 1.26i)9-s + (−0.398 + 1.74i)10-s + (−1.79 − 0.864i)11-s + (0.215 − 0.269i)12-s + (−3.26 − 1.57i)13-s + (0.517 − 0.248i)15-s + (0.0128 − 0.0564i)16-s + (−0.698 + 0.875i)17-s − 2.57·18-s − 7.47·19-s + ⋯
L(s)  = 1  + (−0.561 − 0.270i)2-s + (0.0362 + 0.158i)3-s + (−0.381 − 0.478i)4-s + (−0.202 − 0.886i)5-s + (0.0226 − 0.0990i)6-s + (0.223 + 0.979i)8-s + (0.877 − 0.422i)9-s + (−0.126 + 0.552i)10-s + (−0.541 − 0.260i)11-s + (0.0621 − 0.0779i)12-s + (−0.904 − 0.435i)13-s + (0.133 − 0.0642i)15-s + (0.00321 − 0.0141i)16-s + (−0.169 + 0.212i)17-s − 0.606·18-s − 1.71·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $-0.927 + 0.372i$
Analytic conductor: \(2.73886\)
Root analytic conductor: \(1.65495\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{343} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 343,\ (\ :1/2),\ -0.927 + 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102347 - 0.529287i\)
\(L(\frac12)\) \(\approx\) \(0.102347 - 0.529287i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.794 + 0.382i)T + (1.24 + 1.56i)T^{2} \)
3 \( 1 + (-0.0628 - 0.275i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (0.452 + 1.98i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (1.79 + 0.864i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (3.26 + 1.57i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (0.698 - 0.875i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + 7.47T + 19T^{2} \)
23 \( 1 + (3.31 + 4.16i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-1.28 + 1.60i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + 4.96T + 31T^{2} \)
37 \( 1 + (0.513 - 0.644i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (1.33 + 5.83i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.518 + 2.27i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-4.90 - 2.36i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-8.50 - 10.6i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-2.25 + 9.86i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (5.57 - 6.99i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 + (2.62 + 3.28i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-10.6 + 5.11i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 - 8.26T + 79T^{2} \)
83 \( 1 + (-8.17 + 3.93i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-15.8 + 7.62i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 - 3.72T + 97T^{2} \)
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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

−10.59881196218652185472724844724, −10.36321014026019646177395749655, −9.171943880139279869417105636423, −8.603304432412776612670563966604, −7.60087494016540039193490830436, −6.14809392919631080649207870292, −4.93859682664787372248502884939, −4.19917350485215622352298121053, −2.12999375546796742278779731748, −0.44092790445476578898791075833, 2.26030507408266184406924441769, 3.79936521323518928235085892533, 4.84267448810797538254259267656, 6.62282443184527362771116812490, 7.29784215326377298729680061371, 7.978156933988531704550486878480, 9.121610342114115298098893353196, 10.08194283245824222100300976798, 10.71697830105437270275891096905, 11.98473843362945125019234990775

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