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Properties

Label 2-7e3-7.5-c2-0-10
Degree $2$
Conductor $343$
Sign $-0.500 - 0.866i$
Analytic cond. $9.34607$
Root an. cond. $3.05713$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0990 + 0.171i)2-s + (1.98 + 3.43i)4-s − 1.57·8-s + (−4.5 + 7.79i)9-s + (1.88 + 3.27i)11-s + (−7.76 + 13.4i)16-s + (−0.891 − 1.54i)18-s − 0.748·22-s + (−18.6 + 32.2i)23-s + (−12.5 − 21.6i)25-s + 32.6·29-s + (−4.69 − 8.12i)32-s − 35.6·36-s + (−30.8 + 53.5i)37-s − 13.4·43-s + (−7.48 + 12.9i)44-s + ⋯
L(s)  = 1  + (−0.0495 + 0.0857i)2-s + (0.495 + 0.857i)4-s − 0.197·8-s + (−0.5 + 0.866i)9-s + (0.171 + 0.297i)11-s + (−0.485 + 0.840i)16-s + (−0.0495 − 0.0857i)18-s − 0.0340·22-s + (−0.810 + 1.40i)23-s + (−0.5 − 0.866i)25-s + 1.12·29-s + (−0.146 − 0.253i)32-s − 0.990·36-s + (−0.834 + 1.44i)37-s − 0.313·43-s + (−0.170 + 0.294i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $-0.500 - 0.866i$
Analytic conductor: \(9.34607\)
Root analytic conductor: \(3.05713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{343} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 343,\ (\ :1),\ -0.500 - 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.701243 + 1.21458i\)
\(L(\frac12)\) \(\approx\) \(0.701243 + 1.21458i\)
\(L(2)\) 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 \( 1 + (0.0990 - 0.171i)T + (-2 - 3.46i)T^{2} \)
3 \( 1 + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-1.88 - 3.27i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 169T^{2} \)
17 \( 1 + (144.5 - 250. i)T^{2} \)
19 \( 1 + (180.5 + 312. i)T^{2} \)
23 \( 1 + (18.6 - 32.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 32.6T + 841T^{2} \)
31 \( 1 + (480.5 - 832. i)T^{2} \)
37 \( 1 + (30.8 - 53.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 13.4T + 1.84e3T^{2} \)
47 \( 1 + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-43.2 - 74.8i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-17.8 - 30.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 137.T + 5.04e3T^{2} \)
73 \( 1 + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-78.9 + 136. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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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

−11.76702952368102064680148318991, −10.78575918037660192046231758009, −9.814446448411505037073590949062, −8.519645423111834097145084065160, −7.920257045841612100300626250823, −6.95072984340151257727930296118, −5.85759480473364123860900768585, −4.53273319971038219625132253330, −3.24327030169898240212176224505, −2.03363134234611051137110981258, 0.62988614907741686068486473568, 2.26603639773856183704979709472, 3.68121496212693693890074916247, 5.20375640925127695713119434158, 6.18169634229276997803103168817, 6.88479882672273423004159642122, 8.306600782422120407072390668732, 9.247186953061757015427525674768, 10.12056232913325923075638115963, 10.99111602493581446550121019476

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