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Properties

Label 1-19e2-361.177-r0-0-0
Degree $1$
Conductor $361$
Sign $0.999 + 0.0174i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
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  + (0.919 − 0.393i)2-s + (0.933 + 0.359i)3-s + (0.690 − 0.723i)4-s + (0.888 + 0.459i)5-s + (0.999 − 0.0367i)6-s + (−0.592 + 0.805i)7-s + (0.350 − 0.936i)8-s + (0.741 + 0.670i)9-s + (0.997 + 0.0734i)10-s + (−0.926 + 0.376i)11-s + (0.904 − 0.426i)12-s + (0.515 − 0.856i)13-s + (−0.227 + 0.973i)14-s + (0.663 + 0.748i)15-s + (−0.0459 − 0.998i)16-s + (−0.971 + 0.236i)17-s + ⋯
L(s)  = 1  + (0.919 − 0.393i)2-s + (0.933 + 0.359i)3-s + (0.690 − 0.723i)4-s + (0.888 + 0.459i)5-s + (0.999 − 0.0367i)6-s + (−0.592 + 0.805i)7-s + (0.350 − 0.936i)8-s + (0.741 + 0.670i)9-s + (0.997 + 0.0734i)10-s + (−0.926 + 0.376i)11-s + (0.904 − 0.426i)12-s + (0.515 − 0.856i)13-s + (−0.227 + 0.973i)14-s + (0.663 + 0.748i)15-s + (−0.0459 − 0.998i)16-s + (−0.971 + 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.999 + 0.0174i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.999 + 0.0174i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.169681542 + 0.02758476353i\)
\(L(\frac12)\) \(\approx\) \(3.169681542 + 0.02758476353i\)
\(L(1)\) \(\approx\) \(2.389530517 - 0.06594611308i\)
\(L(1)\) \(\approx\) \(2.389530517 - 0.06594611308i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.919 - 0.393i)T \)
3 \( 1 + (0.933 + 0.359i)T \)
5 \( 1 + (0.888 + 0.459i)T \)
7 \( 1 + (-0.592 + 0.805i)T \)
11 \( 1 + (-0.926 + 0.376i)T \)
13 \( 1 + (0.515 - 0.856i)T \)
17 \( 1 + (-0.971 + 0.236i)T \)
23 \( 1 + (-0.777 - 0.628i)T \)
29 \( 1 + (0.832 - 0.554i)T \)
31 \( 1 + (-0.998 - 0.0550i)T \)
37 \( 1 + (0.789 + 0.614i)T \)
41 \( 1 + (-0.367 - 0.929i)T \)
43 \( 1 + (-0.562 - 0.826i)T \)
47 \( 1 + (-0.467 + 0.883i)T \)
53 \( 1 + (-0.531 - 0.847i)T \)
59 \( 1 + (-0.621 + 0.783i)T \)
61 \( 1 + (0.870 + 0.492i)T \)
67 \( 1 + (-0.333 - 0.942i)T \)
71 \( 1 + (0.870 - 0.492i)T \)
73 \( 1 + (0.280 - 0.959i)T \)
79 \( 1 + (-0.435 + 0.900i)T \)
83 \( 1 + (0.975 + 0.218i)T \)
89 \( 1 + (0.280 + 0.959i)T \)
97 \( 1 + (-0.333 + 0.942i)T \)
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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

−24.72299183408272122551152645835, −23.7807487350318874690404456886, −23.37094847415035697825453354186, −21.778684409344904233779001584856, −21.41734332726820823839853010413, −20.273994631503444843590653218646, −19.947518201470020237200157283763, −18.48002030258014415984458964735, −17.5716710779717616726200453895, −16.31223954233531753335515726726, −15.903982505931316922893156161318, −14.53058285911220688013461123664, −13.74100848126996356143121264964, −13.29440142543784790785192371108, −12.65110430844065677689528293855, −11.20594793737983258550139507685, −9.98227069023495091018424700407, −8.92231466785214113657949350935, −7.9450950325595023047624291865, −6.85989587025330869144507026097, −6.14926381838320813728954593547, −4.797995927843802766129766161860, −3.76897599130333377437313554102, −2.70216863346064478246150540440, −1.63673710470032587805109485497, 2.01939957345955419512264455521, 2.600685403352464018740033787146, 3.50618528994721433358846482651, 4.8450978092805587704509881624, 5.82839848529552544137049710091, 6.7607803810491739782642467558, 8.16249964725993015851287455039, 9.39039305634896195857704968246, 10.20571692132291833481673544782, 10.85201119793767613408650749530, 12.44705110826257815691589506595, 13.19047653302012408338255731098, 13.76357620576125451745583746745, 14.92788884470501774223948499722, 15.40335483352761852399431626464, 16.23292373542001147545085543471, 18.00999062430780198154238445343, 18.66769172607273120447572940722, 19.715909214522553093849718098556, 20.55115229812691166378636347466, 21.223619137846537956264686428013, 22.12743982895012825882375127352, 22.5100969322766115804438847470, 23.87582962877658851910689997495, 24.86992663671380223250270389694

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