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Properties

Label 2-329-329.93-c0-0-1
Degree $2$
Conductor $329$
Sign $-0.784 - 0.620i$
Analytic cond. $0.164192$
Root an. cond. $0.405206$
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  + (−0.669 + 1.15i)2-s + (0.809 + 1.40i)3-s + (−0.395 − 0.684i)4-s − 2.16·6-s + (0.669 − 0.743i)7-s − 0.279·8-s + (−0.809 + 1.40i)9-s + (0.639 − 1.10i)12-s + (0.413 + 1.27i)14-s + (0.582 − 1.00i)16-s + (−0.913 − 1.58i)17-s + (−1.08 − 1.87i)18-s + (1.58 + 0.336i)21-s + (−0.226 − 0.392i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.669 + 1.15i)2-s + (0.809 + 1.40i)3-s + (−0.395 − 0.684i)4-s − 2.16·6-s + (0.669 − 0.743i)7-s − 0.279·8-s + (−0.809 + 1.40i)9-s + (0.639 − 1.10i)12-s + (0.413 + 1.27i)14-s + (0.582 − 1.00i)16-s + (−0.913 − 1.58i)17-s + (−1.08 − 1.87i)18-s + (1.58 + 0.336i)21-s + (−0.226 − 0.392i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 329 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 329 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(329\)    =    \(7 \cdot 47\)
Sign: $-0.784 - 0.620i$
Analytic conductor: \(0.164192\)
Root analytic conductor: \(0.405206\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{329} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 329,\ (\ :0),\ -0.784 - 0.620i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7333855700\)
\(L(\frac12)\) \(\approx\) \(0.7333855700\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.669 + 0.743i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.669 - 1.15i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.913 + 1.58i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.913 - 1.58i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
53 \( 1 + (-0.978 - 1.69i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.669 + 1.15i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.669 - 1.15i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.82T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.95T + T^{2} \)
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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

−11.91438539982107099670360261650, −10.87196333030955357408634184581, −9.948483395748035623363736136830, −9.236255472473650207245477498343, −8.449914372338562942614672988005, −7.67625808944133744472150545490, −6.65558505077137405488207586256, −5.13171704548076107549426400459, −4.31284991009843045661816457700, −2.90183104402826162231304474382, 1.68645756168233095062941011887, 2.26449363185006760463762289466, 3.64692453493006510428542000412, 5.71329851781330750415844131549, 6.85407118795595001850969981394, 8.125258701577221978287428657344, 8.588071381174095460277208669792, 9.409601448047303321602950053638, 10.72225150387065585578446362631, 11.47569131598313690268391917731

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