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Properties

Label 2-399-7.4-c1-0-2
Degree $2$
Conductor $399$
Sign $0.968 + 0.250i$
Analytic cond. $3.18603$
Root an. cond. $1.78494$
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  + (−1 − 1.73i)2-s + (−0.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + (−0.5 − 0.866i)5-s + 1.99·6-s + (2 + 1.73i)7-s + (−0.499 − 0.866i)9-s + (−0.999 + 1.73i)10-s + (−2 + 3.46i)11-s + (−1 − 1.73i)12-s + 4·13-s + (0.999 − 5.19i)14-s + 0.999·15-s + (1.99 + 3.46i)16-s + (−1.5 + 2.59i)17-s + (−0.999 + 1.73i)18-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.288 + 0.499i)3-s + (−0.499 + 0.866i)4-s + (−0.223 − 0.387i)5-s + 0.816·6-s + (0.755 + 0.654i)7-s + (−0.166 − 0.288i)9-s + (−0.316 + 0.547i)10-s + (−0.603 + 1.04i)11-s + (−0.288 − 0.499i)12-s + 1.10·13-s + (0.267 − 1.38i)14-s + 0.258·15-s + (0.499 + 0.866i)16-s + (−0.363 + 0.630i)17-s + (−0.235 + 0.408i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(3.18603\)
Root analytic conductor: \(1.78494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.807395 - 0.102889i\)
\(L(\frac12)\) \(\approx\) \(0.807395 - 0.102889i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2 - 1.73i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (1 + 1.73i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (3 - 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−11.11434093392526582166926013506, −10.43055393104856466330457629662, −9.652349723397123301989287590201, −8.593234665854215871497201625291, −8.184251692188129543680124919564, −6.41309874332938877159518966720, −5.15765786481309586860098007497, −4.14527672094507181722081981980, −2.71305702172341658616408371405, −1.37748747902945203765597400910, 0.807058801277600837067674566414, 3.07963926922204966930739485640, 4.82702870174306281283561689745, 5.96042093331962637230411236484, 6.75918380297661547156631849617, 7.59655431279357530699910183427, 8.300270065100681228671581932739, 9.025644260089397023129824592118, 10.56247610141483054139434048144, 11.06082717898497557684684798234

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