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Properties

Label 1-1339-1339.1082-r0-0-0
Degree $1$
Conductor $1339$
Sign $-0.372 + 0.928i$
Analytic cond. $6.21828$
Root an. cond. $6.21828$
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  + (−0.602 + 0.798i)2-s + (−0.998 − 0.0615i)3-s + (−0.273 − 0.961i)4-s + (−0.908 − 0.417i)5-s + (0.650 − 0.759i)6-s + (0.932 − 0.361i)7-s + (0.932 + 0.361i)8-s + (0.992 + 0.122i)9-s + (0.881 − 0.473i)10-s + (0.992 + 0.122i)11-s + (0.213 + 0.976i)12-s + (−0.273 + 0.961i)14-s + (0.881 + 0.473i)15-s + (−0.850 + 0.526i)16-s + (−0.982 + 0.183i)17-s + (−0.696 + 0.717i)18-s + ⋯
L(s)  = 1  + (−0.602 + 0.798i)2-s + (−0.998 − 0.0615i)3-s + (−0.273 − 0.961i)4-s + (−0.908 − 0.417i)5-s + (0.650 − 0.759i)6-s + (0.932 − 0.361i)7-s + (0.932 + 0.361i)8-s + (0.992 + 0.122i)9-s + (0.881 − 0.473i)10-s + (0.992 + 0.122i)11-s + (0.213 + 0.976i)12-s + (−0.273 + 0.961i)14-s + (0.881 + 0.473i)15-s + (−0.850 + 0.526i)16-s + (−0.982 + 0.183i)17-s + (−0.696 + 0.717i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $-0.372 + 0.928i$
Analytic conductor: \(6.21828\)
Root analytic conductor: \(6.21828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (1082, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1339,\ (0:\ ),\ -0.372 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3207072303 + 0.4742927657i\)
\(L(\frac12)\) \(\approx\) \(0.3207072303 + 0.4742927657i\)
\(L(1)\) \(\approx\) \(0.5295498037 + 0.1853253217i\)
\(L(1)\) \(\approx\) \(0.5295498037 + 0.1853253217i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.602 + 0.798i)T \)
3 \( 1 + (-0.998 - 0.0615i)T \)
5 \( 1 + (-0.908 - 0.417i)T \)
7 \( 1 + (0.932 - 0.361i)T \)
11 \( 1 + (0.992 + 0.122i)T \)
17 \( 1 + (-0.982 + 0.183i)T \)
19 \( 1 + (0.552 + 0.833i)T \)
23 \( 1 + (0.992 - 0.122i)T \)
29 \( 1 + (0.0922 + 0.995i)T \)
31 \( 1 + (-0.850 + 0.526i)T \)
37 \( 1 + (0.213 + 0.976i)T \)
41 \( 1 + (0.0922 - 0.995i)T \)
43 \( 1 + (-0.952 + 0.303i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.552 + 0.833i)T \)
59 \( 1 + (-0.779 + 0.626i)T \)
61 \( 1 + (0.650 + 0.759i)T \)
67 \( 1 + (-0.153 - 0.988i)T \)
71 \( 1 + (0.0922 - 0.995i)T \)
73 \( 1 + (0.0922 - 0.995i)T \)
79 \( 1 + (0.0922 + 0.995i)T \)
83 \( 1 + (-0.779 - 0.626i)T \)
89 \( 1 + (-0.696 - 0.717i)T \)
97 \( 1 + (0.650 - 0.759i)T \)
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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

−20.55731871612827877750683399492, −19.85286531857317298949872932242, −19.08092872741925926873241827314, −18.35337461745439623214574941076, −17.76026109133990322406764886747, −17.11907271321455590329847947638, −16.30425923763718123862692683516, −15.46741738567024401141142437287, −14.72105855876180723478565375337, −13.49914336144427322891626184942, −12.635071728446184873164767350891, −11.69034212915053362967127510316, −11.32566169480299100924609339253, −11.06390315259635546028860919110, −9.85712033052237746231255311567, −9.02559586829372626604081650121, −8.194712068848020184557382204859, −7.21297930551866592412632739006, −6.69184552745643717013062329644, −5.25755338836922371390429604077, −4.41693420632479185895288278561, −3.77755900714779103614001560589, −2.54621245859976366368070667416, −1.4612200538369581126493327898, −0.403886703757925063282809625821, 1.08446814963387950855352806270, 1.57031033508041695801056761431, 3.757229633536466738253371899423, 4.6548945567963280081616750231, 5.08565678169645510556335236410, 6.20612376629089081164373363773, 7.06586271977964760863308963744, 7.5548884229946540390187999749, 8.56643401022776927312638185354, 9.2051662770906549804802175554, 10.410753557631078161952907710, 11.0473588626427985624823946669, 11.67275902504465751297068687921, 12.52279726119057951730406063233, 13.570023117517698704072452874647, 14.583289770775347903258557816, 15.18596640745593411111359326266, 16.01866778778858413942908994203, 16.80276427227888958272069600480, 17.08196511855308453657198049633, 18.05084997588260602354427759837, 18.53762177991542311021210258979, 19.59287808064942447417463653703, 20.10973791978366773778022259872, 21.08966392584958896260421382308

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