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Properties

Label 1-3381-3381.629-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.754 - 0.656i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (−0.488 + 0.872i)2-s + (−0.523 − 0.852i)4-s + (0.685 + 0.728i)5-s + (0.999 − 0.0407i)8-s + (−0.970 + 0.242i)10-s + (0.0611 + 0.998i)11-s + (−0.794 − 0.607i)13-s + (−0.452 + 0.891i)16-s + (−0.523 + 0.852i)17-s + (−0.841 − 0.540i)19-s + (0.262 − 0.965i)20-s + (−0.900 − 0.433i)22-s + (−0.0611 + 0.998i)25-s + (0.917 − 0.396i)26-s + (0.523 − 0.852i)29-s + ⋯
L(s)  = 1  + (−0.488 + 0.872i)2-s + (−0.523 − 0.852i)4-s + (0.685 + 0.728i)5-s + (0.999 − 0.0407i)8-s + (−0.970 + 0.242i)10-s + (0.0611 + 0.998i)11-s + (−0.794 − 0.607i)13-s + (−0.452 + 0.891i)16-s + (−0.523 + 0.852i)17-s + (−0.841 − 0.540i)19-s + (0.262 − 0.965i)20-s + (−0.900 − 0.433i)22-s + (−0.0611 + 0.998i)25-s + (0.917 − 0.396i)26-s + (0.523 − 0.852i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.754 - 0.656i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.754 - 0.656i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2052666758 + 0.5488279861i\)
\(L(\frac12)\) \(\approx\) \(-0.2052666758 + 0.5488279861i\)
\(L(1)\) \(\approx\) \(0.6050211455 + 0.4443707098i\)
\(L(1)\) \(\approx\) \(0.6050211455 + 0.4443707098i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.488 + 0.872i)T \)
5 \( 1 + (0.685 + 0.728i)T \)
11 \( 1 + (0.0611 + 0.998i)T \)
13 \( 1 + (-0.794 - 0.607i)T \)
17 \( 1 + (-0.523 + 0.852i)T \)
19 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (0.523 - 0.852i)T \)
31 \( 1 + (0.654 + 0.755i)T \)
37 \( 1 + (0.557 + 0.830i)T \)
41 \( 1 + (0.685 + 0.728i)T \)
43 \( 1 + (-0.999 - 0.0407i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (-0.0203 + 0.999i)T \)
59 \( 1 + (0.970 - 0.242i)T \)
61 \( 1 + (-0.917 - 0.396i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.742 - 0.670i)T \)
73 \( 1 + (-0.339 + 0.940i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (-0.933 + 0.359i)T \)
89 \( 1 + (0.882 + 0.470i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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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

−18.40830085368833068194347499049, −17.785519774434483682837961735, −17.07726327115900825301487460807, −16.48339273904852089168699293845, −16.08384364478737922688883811364, −14.70872693915542716070176783144, −14.00459612739168355870875336526, −13.37129396539113831203465192858, −12.8124942044676162049231113019, −11.95456633697162300469030293926, −11.51085932799152325213960650745, −10.53448420946439123882264085324, −9.99173511936346649725779834883, −9.128440445219416674044724030999, −8.78906611227079920690196921654, −8.01069821830748902713617424186, −7.05880121667160825296292258847, −6.17039937906642188407277785417, −5.199473435307492290707836974563, −4.53006459274308659097193892158, −3.74890237530205796363176438041, −2.64626296164414630518520875289, −2.116344992931822224283151264458, −1.16008345025993060031464339612, −0.21075351705569833488916004443, 1.32833701599501415542740872496, 2.17413144073579618750715616103, 2.96474946528506360621999963630, 4.41728623965800289265418361497, 4.785481095377900674435435690062, 5.92853611206890943641021108861, 6.40248050204988945650736941093, 7.075910510729900041501785797642, 7.79542802388219541448567427843, 8.556984243961701316397229518841, 9.44714138194064653489510323425, 10.05312137835332072057719347596, 10.480668762086310487870790166294, 11.32225574879093199684360677773, 12.48292578156543713122345200121, 13.159366181878400641757715360397, 13.83948677192322065525249044429, 14.72955652249456821934115080788, 15.06530866991825474420004390418, 15.5746249625430728860966568821, 16.70034176977965873278270312762, 17.41203566527068129234998185778, 17.62324222078156528710058267398, 18.28130869383680179914676890736, 19.25437317817786618935197442255

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