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Properties

Label 1-7e3-343.33-r1-0-0
Degree $1$
Conductor $343$
Sign $-0.204 + 0.978i$
Analytic cond. $36.8604$
Root an. cond. $36.8604$
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  + (0.849 − 0.527i)2-s + (−0.243 − 0.969i)3-s + (0.443 − 0.896i)4-s + (−0.747 − 0.664i)5-s + (−0.718 − 0.695i)6-s + (−0.0960 − 0.995i)8-s + (−0.881 + 0.471i)9-s + (−0.985 − 0.170i)10-s + (0.656 − 0.754i)11-s + (−0.977 − 0.212i)12-s + (−0.801 − 0.598i)13-s + (−0.462 + 0.886i)15-s + (−0.606 − 0.794i)16-s + (0.138 − 0.990i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.849 − 0.527i)2-s + (−0.243 − 0.969i)3-s + (0.443 − 0.896i)4-s + (−0.747 − 0.664i)5-s + (−0.718 − 0.695i)6-s + (−0.0960 − 0.995i)8-s + (−0.881 + 0.471i)9-s + (−0.985 − 0.170i)10-s + (0.656 − 0.754i)11-s + (−0.977 − 0.212i)12-s + (−0.801 − 0.598i)13-s + (−0.462 + 0.886i)15-s + (−0.606 − 0.794i)16-s + (0.138 − 0.990i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(36.8604\)
Root analytic conductor: \(36.8604\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{343} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 343,\ (1:\ ),\ -0.204 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9508456784 - 1.170173491i\)
\(L(\frac12)\) \(\approx\) \(-0.9508456784 - 1.170173491i\)
\(L(1)\) \(\approx\) \(0.6536178337 - 1.032679031i\)
\(L(1)\) \(\approx\) \(0.6536178337 - 1.032679031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (0.849 - 0.527i)T \)
3 \( 1 + (-0.243 - 0.969i)T \)
5 \( 1 + (-0.747 - 0.664i)T \)
11 \( 1 + (0.656 - 0.754i)T \)
13 \( 1 + (-0.801 - 0.598i)T \)
17 \( 1 + (0.138 - 0.990i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.860 - 0.509i)T \)
29 \( 1 + (0.871 + 0.490i)T \)
31 \( 1 + (-0.955 - 0.294i)T \)
37 \( 1 + (-0.788 - 0.615i)T \)
41 \( 1 + (0.838 + 0.545i)T \)
43 \( 1 + (-0.0960 + 0.995i)T \)
47 \( 1 + (0.536 - 0.843i)T \)
53 \( 1 + (0.775 + 0.631i)T \)
59 \( 1 + (0.0534 + 0.998i)T \)
61 \( 1 + (0.972 - 0.232i)T \)
67 \( 1 + (-0.733 - 0.680i)T \)
71 \( 1 + (0.871 - 0.490i)T \)
73 \( 1 + (0.536 + 0.843i)T \)
79 \( 1 + (0.365 - 0.930i)T \)
83 \( 1 + (-0.967 - 0.253i)T \)
89 \( 1 + (0.180 - 0.983i)T \)
97 \( 1 + (0.222 + 0.974i)T \)
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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

−25.48713685461100768077157684317, −24.05509536334115718303393065076, −23.53996111875141516150601513233, −22.36726575187191694815584190913, −22.193056589521165647976665379451, −21.237200999722927384165278615912, −20.088163792727030192869860489984, −19.459675352231780151622191771564, −17.73775115232690302079060817368, −17.10945638450070510691021795403, −16.01981466608013724178784151907, −15.38445010377482053693520972296, −14.6374524084894611550400271828, −13.99125461225777433209227832031, −12.329034621941448775046607821555, −11.792531092908720498003799058559, −10.85488087133006864666994238667, −9.722186037971663319351855258377, −8.502640503301022190330327411002, −7.29987749013823402582218903504, −6.514639323686733703070272711258, −5.28380924652788809652629068457, −4.24309820372466703713011260879, −3.67513787396450910980543563958, −2.396152015841182127043108471770, 0.34410514006228738682598453204, 1.26298560722271915670899321075, 2.66441709705412979036251234728, 3.78025049816166798214599220357, 5.054361674687687867124156857486, 5.849852569260250348652344861333, 7.05746241096728728114886420583, 7.96932627937593385005066347684, 9.21506794851327459902319764248, 10.599139736572218882781472981313, 11.73478562392923346862241821816, 12.103323710465941369799008338458, 12.92479327831468023031981166840, 13.96349446186201828227696729487, 14.63569164105532623308979008075, 16.045532943397334304815516219078, 16.65114518540237817581445187363, 18.06737278319208870776125614005, 18.93575243728937819603053860716, 19.884011368347179021647328800192, 20.16269813785438732966560795552, 21.48413123522185223768257525660, 22.60430752074724887720129257243, 23.00920237896008533204689925819, 24.15796704379919780463285800441

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