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Properties

Label 2-342-171.110-c1-0-9
Degree $2$
Conductor $342$
Sign $0.996 + 0.0873i$
Analytic cond. $2.73088$
Root an. cond. $1.65253$
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  + (0.766 − 0.642i)2-s + (1.60 − 0.647i)3-s + (0.173 − 0.984i)4-s + (−1.41 + 3.88i)5-s + (0.814 − 1.52i)6-s + (2.04 + 3.54i)7-s + (−0.500 − 0.866i)8-s + (2.16 − 2.08i)9-s + (1.41 + 3.88i)10-s − 4.97i·11-s + (−0.358 − 1.69i)12-s + (0.452 + 1.24i)13-s + (3.84 + 1.40i)14-s + (0.244 + 7.15i)15-s + (−0.939 − 0.342i)16-s + (1.10 − 3.03i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (0.927 − 0.373i)3-s + (0.0868 − 0.492i)4-s + (−0.632 + 1.73i)5-s + (0.332 − 0.624i)6-s + (0.773 + 1.34i)7-s + (−0.176 − 0.306i)8-s + (0.720 − 0.693i)9-s + (0.447 + 1.22i)10-s − 1.50i·11-s + (−0.103 − 0.489i)12-s + (0.125 + 0.344i)13-s + (1.02 + 0.374i)14-s + (0.0630 + 1.84i)15-s + (−0.234 − 0.0855i)16-s + (0.267 − 0.736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.996 + 0.0873i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.996 + 0.0873i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24556 - 0.0982954i\)
\(L(\frac12)\) \(\approx\) \(2.24556 - 0.0982954i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (-1.60 + 0.647i)T \)
19 \( 1 + (1.63 - 4.04i)T \)
good5 \( 1 + (1.41 - 3.88i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-2.04 - 3.54i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.97iT - 11T^{2} \)
13 \( 1 + (-0.452 - 1.24i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.10 + 3.03i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (3.60 + 0.635i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.14 + 6.49i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 - 1.67iT - 31T^{2} \)
37 \( 1 + 0.879iT - 37T^{2} \)
41 \( 1 + (-0.319 + 0.267i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.381 + 2.16i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (8.12 + 1.43i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-1.78 - 1.49i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-1.22 - 6.95i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-5.46 + 1.99i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-4.68 + 5.58i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (5.35 - 4.49i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.522 + 2.96i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (3.63 - 9.98i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-7.01 + 4.05i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.49 - 8.46i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-5.33 - 6.36i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.68338391488979788542931612598, −10.84534029761003292822480935243, −9.779813533443670043663048508457, −8.499462608215301509800669986742, −7.88240886507269683710583481317, −6.60834996155968111402228322854, −5.77840550807635937457070502331, −3.94509731519260667990239431326, −3.02336421712130847066802518557, −2.20397319917719844148701555297, 1.60405785314943197950273721817, 3.80944501266266069854090966406, 4.50729342404061062711722365941, 5.02796544567707602053790103811, 7.09630571193661906816131703474, 7.88656751228053842472242681722, 8.426901102610431223567444361657, 9.502596109525352020523559663615, 10.52050753759285045520066425740, 11.76858928213378856046804748947

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