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Properties

Label 1-3381-3381.3194-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.530 - 0.847i$
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  + (0.352 + 0.935i)2-s + (−0.751 + 0.659i)4-s + (0.777 + 0.628i)5-s + (−0.882 − 0.470i)8-s + (−0.314 + 0.949i)10-s + (−0.951 + 0.307i)11-s + (0.0203 − 0.999i)13-s + (0.128 − 0.991i)16-s + (−0.195 + 0.980i)17-s + (0.888 + 0.458i)19-s + (−0.999 + 0.0407i)20-s + (−0.623 − 0.781i)22-s + (0.209 + 0.977i)25-s + (0.942 − 0.333i)26-s + (−0.947 + 0.320i)29-s + ⋯
L(s)  = 1  + (0.352 + 0.935i)2-s + (−0.751 + 0.659i)4-s + (0.777 + 0.628i)5-s + (−0.882 − 0.470i)8-s + (−0.314 + 0.949i)10-s + (−0.951 + 0.307i)11-s + (0.0203 − 0.999i)13-s + (0.128 − 0.991i)16-s + (−0.195 + 0.980i)17-s + (0.888 + 0.458i)19-s + (−0.999 + 0.0407i)20-s + (−0.623 − 0.781i)22-s + (0.209 + 0.977i)25-s + (0.942 − 0.333i)26-s + (−0.947 + 0.320i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.530 - 0.847i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (3194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.530 - 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4075787941 + 0.7360991057i\)
\(L(\frac12)\) \(\approx\) \(-0.4075787941 + 0.7360991057i\)
\(L(1)\) \(\approx\) \(0.7705223146 + 0.7119383527i\)
\(L(1)\) \(\approx\) \(0.7705223146 + 0.7119383527i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.352 + 0.935i)T \)
5 \( 1 + (0.777 + 0.628i)T \)
11 \( 1 + (-0.951 + 0.307i)T \)
13 \( 1 + (0.0203 - 0.999i)T \)
17 \( 1 + (-0.195 + 0.980i)T \)
19 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (-0.947 + 0.320i)T \)
31 \( 1 + (-0.327 - 0.945i)T \)
37 \( 1 + (-0.288 + 0.957i)T \)
41 \( 1 + (0.933 - 0.359i)T \)
43 \( 1 + (-0.882 + 0.470i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (-0.275 + 0.961i)T \)
59 \( 1 + (-0.314 + 0.949i)T \)
61 \( 1 + (-0.760 + 0.649i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (0.818 + 0.574i)T \)
73 \( 1 + (-0.476 - 0.879i)T \)
79 \( 1 + (-0.580 + 0.814i)T \)
83 \( 1 + (-0.301 - 0.953i)T \)
89 \( 1 + (-0.115 - 0.993i)T \)
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

−18.31761820203405306113911050907, −17.976491288605411867799812197113, −17.115861758882503760786034286189, −16.19768046526598239005898489533, −15.733058383738051940988418584738, −14.55587735152982837156979099736, −13.91544095064638516895884279826, −13.51641108400257758890350776021, −12.785702831669897261785507335980, −12.17059351372308420561298189115, −11.27798403002371676872925710125, −10.840113195583575123838057714116, −9.7672725060714705851501680918, −9.40322898834836298125649505760, −8.78521024832118159374519109716, −7.808405006709169693208038878158, −6.76893876448452649023926664417, −5.84682541868397712070633445951, −5.11693676494662045667502075990, −4.74547824079726164874521863138, −3.66809180837056718114540355697, −2.788275818181339268808284881848, −2.084709882582917680515752309300, −1.30421630988220345015737616776, −0.2089291442418241917195396141, 1.42989034359528022396211987588, 2.58869995719958574277134502872, 3.228659493802900932431437501174, 4.102292466064039547748262836434, 5.18972902407279144298034612400, 5.66918883590200657814620174159, 6.228813206910594425635750395414, 7.2545091625909024034507494379, 7.682800642959128759235729222599, 8.477963216358485352004437231540, 9.39242094302560016286443025225, 10.11442153925478018299802741328, 10.65111723705456807909987759911, 11.70386117282835680536418248599, 12.70338663559683137256335455435, 13.17147993996883024452065205504, 13.6989052122798571033165861816, 14.63029239722968256518470223130, 15.084946518380467341042240047400, 15.62525553007793031145519853493, 16.57278416700471593789205921206, 17.16906721931213050852666884378, 17.878004640000357866409954846749, 18.34273341145529017589296999927, 18.88458154318508708702692287329

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