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Properties

Label 2-342-9.7-c1-0-9
Degree $2$
Conductor $342$
Sign $0.446 - 0.894i$
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  + (−0.5 + 0.866i)2-s + (1.71 + 0.248i)3-s + (−0.499 − 0.866i)4-s + (2.19 + 3.80i)5-s + (−1.07 + 1.36i)6-s + (1.88 − 3.26i)7-s + 0.999·8-s + (2.87 + 0.852i)9-s − 4.38·10-s + (0.852 − 1.47i)11-s + (−0.641 − 1.60i)12-s + (−2.01 − 3.48i)13-s + (1.88 + 3.26i)14-s + (2.81 + 7.05i)15-s + (−0.5 + 0.866i)16-s − 7.47·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.989 + 0.143i)3-s + (−0.249 − 0.433i)4-s + (0.981 + 1.69i)5-s + (−0.437 + 0.555i)6-s + (0.711 − 1.23i)7-s + 0.353·8-s + (0.958 + 0.284i)9-s − 1.38·10-s + (0.256 − 0.445i)11-s + (−0.185 − 0.464i)12-s + (−0.558 − 0.967i)13-s + (0.503 + 0.871i)14-s + (0.727 + 1.82i)15-s + (−0.125 + 0.216i)16-s − 1.81·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.446 - 0.894i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.446 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51509 + 0.937517i\)
\(L(\frac12)\) \(\approx\) \(1.51509 + 0.937517i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-1.71 - 0.248i)T \)
19 \( 1 + T \)
good5 \( 1 + (-2.19 - 3.80i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.88 + 3.26i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.852 + 1.47i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.01 + 3.48i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.47T + 17T^{2} \)
23 \( 1 + (0.0657 + 0.113i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.592 + 1.02i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.582 - 1.00i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.75T + 37T^{2} \)
41 \( 1 + (2.65 + 4.60i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.740 - 1.28i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.38 - 9.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.341T + 53T^{2} \)
59 \( 1 + (-0.328 - 0.568i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.80 - 4.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.84 + 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 - 6.40T + 73T^{2} \)
79 \( 1 + (-1.33 + 2.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.879 + 1.52i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + (-6.88 + 11.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.07442376532402652948213551087, −10.56602759398137417703773035556, −9.920986743265919277126130728811, −8.859645358439344201446718599479, −7.71069771821971895145121805196, −7.08475360995489080388149042138, −6.24716194970414219797047394358, −4.65119848740785299347404456068, −3.28180453425924446196157731880, −2.00317346764932069177394924670, 1.80277602606508510427751121571, 2.19591752694354237537511820657, 4.37450537874642111248166627316, 5.03214338005937376007814801516, 6.60890557893036175480474577181, 8.183634617322202069035747312629, 8.922815223674821662139363735678, 9.123317405467524857481918986987, 10.04850385412715004275338948571, 11.65178790352695753243855669595

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