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Properties

Label 2-342-171.110-c1-0-7
Degree $2$
Conductor $342$
Sign $0.984 + 0.176i$
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 + (−1.45 + 0.940i)3-s + (0.173 − 0.984i)4-s + (0.146 − 0.403i)5-s + (0.509 − 1.65i)6-s + (−0.587 − 1.01i)7-s + (0.500 + 0.866i)8-s + (1.23 − 2.73i)9-s + (0.146 + 0.403i)10-s + 2.82i·11-s + (0.673 + 1.59i)12-s + (−1.76 − 4.86i)13-s + (1.10 + 0.401i)14-s + (0.165 + 0.724i)15-s + (−0.939 − 0.342i)16-s + (1.80 − 4.94i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (−0.839 + 0.542i)3-s + (0.0868 − 0.492i)4-s + (0.0656 − 0.180i)5-s + (0.208 − 0.675i)6-s + (−0.221 − 0.384i)7-s + (0.176 + 0.306i)8-s + (0.410 − 0.911i)9-s + (0.0463 + 0.127i)10-s + 0.852i·11-s + (0.194 + 0.460i)12-s + (−0.490 − 1.34i)13-s + (0.294 + 0.107i)14-s + (0.0427 + 0.186i)15-s + (−0.234 − 0.0855i)16-s + (0.436 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.984 + 0.176i$
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: \((2,\ 342,\ (\ :1/2),\ 0.984 + 0.176i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.717148 - 0.0636117i\)
\(L(\frac12)\) \(\approx\) \(0.717148 - 0.0636117i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 - 0.940i)T \)
19 \( 1 + (-3.29 + 2.85i)T \)
good5 \( 1 + (-0.146 + 0.403i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.587 + 1.01i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + (1.76 + 4.86i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.80 + 4.94i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-8.67 - 1.52i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.843 - 4.78i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 - 1.49iT - 31T^{2} \)
37 \( 1 + 7.77iT - 37T^{2} \)
41 \( 1 + (1.25 - 1.05i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (1.90 + 10.7i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-2.72 - 0.480i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (3.74 + 3.14i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (1.20 + 6.81i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (7.06 - 2.57i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-6.67 + 7.95i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.04 - 0.879i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.0657 + 0.372i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (5.85 - 16.0i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (4.48 - 2.58i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.84 + 10.4i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-0.0203 - 0.0242i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.21989795134793695856093869781, −10.48394060334553973531226438105, −9.641510362694153730107249199643, −8.996475654018369765521912984081, −7.29831416499591468113027555191, −7.03004775888597019204775461091, −5.28970601328780421323645283721, −5.06132294543009300359877800971, −3.21865456143013440902827780417, −0.77243537363180573390598922041, 1.34636623219399545823993749143, 2.88916712521452500481432543638, 4.53585631852297026274532629701, 5.92074669199584428728974498607, 6.70870637765742394476183030377, 7.79517774150761037007151917653, 8.801355873237419931138942086549, 9.847343192510884807564315899586, 10.77531423269827289548985616149, 11.52671225148848773870084106291

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