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Properties

Label 1-7e3-343.172-r0-0-0
Degree $1$
Conductor $343$
Sign $0.962 - 0.269i$
Analytic cond. $1.59288$
Root an. cond. $1.59288$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.996 − 0.0853i)2-s + (−0.536 + 0.843i)3-s + (0.985 − 0.170i)4-s + (0.656 − 0.754i)5-s + (−0.462 + 0.886i)6-s + (0.967 − 0.253i)8-s + (−0.424 − 0.905i)9-s + (0.589 − 0.807i)10-s + (−0.0106 − 0.999i)11-s + (−0.385 + 0.922i)12-s + (0.926 − 0.375i)13-s + (0.284 + 0.958i)15-s + (0.942 − 0.335i)16-s + (−0.999 − 0.0213i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.996 − 0.0853i)2-s + (−0.536 + 0.843i)3-s + (0.985 − 0.170i)4-s + (0.656 − 0.754i)5-s + (−0.462 + 0.886i)6-s + (0.967 − 0.253i)8-s + (−0.424 − 0.905i)9-s + (0.589 − 0.807i)10-s + (−0.0106 − 0.999i)11-s + (−0.385 + 0.922i)12-s + (0.926 − 0.375i)13-s + (0.284 + 0.958i)15-s + (0.942 − 0.335i)16-s + (−0.999 − 0.0213i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $0.962 - 0.269i$
Analytic conductor: \(1.59288\)
Root analytic conductor: \(1.59288\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{343} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 343,\ (0:\ ),\ 0.962 - 0.269i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.241931830 - 0.3082222425i\)
\(L(\frac12)\) \(\approx\) \(2.241931830 - 0.3082222425i\)
\(L(1)\) \(\approx\) \(1.792829615 - 0.06890685884i\)
\(L(1)\) \(\approx\) \(1.792829615 - 0.06890685884i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (0.996 - 0.0853i)T \)
3 \( 1 + (-0.536 + 0.843i)T \)
5 \( 1 + (0.656 - 0.754i)T \)
11 \( 1 + (-0.0106 - 0.999i)T \)
13 \( 1 + (0.926 - 0.375i)T \)
17 \( 1 + (-0.999 - 0.0213i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.201 - 0.979i)T \)
29 \( 1 + (-0.949 - 0.315i)T \)
31 \( 1 + (0.0747 + 0.997i)T \)
37 \( 1 + (0.481 + 0.876i)T \)
41 \( 1 + (0.0320 + 0.999i)T \)
43 \( 1 + (0.967 + 0.253i)T \)
47 \( 1 + (0.687 + 0.725i)T \)
53 \( 1 + (-0.814 + 0.580i)T \)
59 \( 1 + (-0.881 + 0.471i)T \)
61 \( 1 + (-0.961 + 0.274i)T \)
67 \( 1 + (0.826 + 0.563i)T \)
71 \( 1 + (-0.949 + 0.315i)T \)
73 \( 1 + (0.687 - 0.725i)T \)
79 \( 1 + (0.955 + 0.294i)T \)
83 \( 1 + (0.159 - 0.987i)T \)
89 \( 1 + (0.977 - 0.212i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.786046441099368047777079611864, −23.87462658233171083808531371900, −23.15814602451751521111101319619, −22.40205895427769485128282431827, −21.75163606539069304473032908393, −20.68049644005673457175350884058, −19.662709907412654209693199767472, −18.6682713862357661164172334583, −17.67012489828217927924690826519, −17.07280151386726633540980605644, −15.730483141213372078301615677374, −14.91575652761730688129170007891, −13.76690644246994744507801635095, −13.29443705047275612646139285163, −12.40176401039864816624905132338, −11.14120432243246930118780131830, −10.88443922909881680799540073978, −9.27631059176081129359825245503, −7.60455258769634488982917222609, −6.86535140599728286430313830638, −6.158648515524756580320689808284, −5.21359473419002625097907379954, −3.94537326744038467828761000061, −2.42082849934982233159937149640, −1.780656886240496251178208770712, 1.22735071073688485996725441468, 2.84651878169155922477270672307, 4.03085455862798788268146924271, 4.83768450578516681980755879074, 5.96584528688448249810488744860, 6.274192339339041349710590757282, 8.245738820404864770402910079904, 9.24359953727731367957010638338, 10.54269021638243313934560171984, 11.03541636702067689830361622955, 12.20509109075039542408228276384, 13.08127066931122628299256528093, 13.92807876743592837458743626044, 14.951430396226641434919597178030, 15.99143597340224447387174715761, 16.48423823975647895213152254856, 17.37027404565901711506958114440, 18.64217067517646875699376223166, 20.099207320795655902724931028430, 20.7374077441284173949602397780, 21.39584793227669546153754874292, 22.127340724527262113050058126609, 22.96404218186708692509571482586, 23.866381756817984151711496003084, 24.66300907903231697701468769791

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