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Properties

Label 2-342-57.50-c1-0-3
Degree $2$
Conductor $342$
Sign $-0.600 + 0.799i$
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 + (−0.499 + 0.866i)4-s + (1.22 − 0.707i)5-s − 3.44·7-s + 0.999·8-s + (−1.22 − 0.707i)10-s − 6.29i·11-s + (−2.17 − 1.25i)13-s + (1.72 + 2.98i)14-s + (−0.5 − 0.866i)16-s + (4.22 − 2.43i)17-s + (4 − 1.73i)19-s + 1.41i·20-s + (−5.44 + 3.14i)22-s + (−4.89 − 2.82i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.547 − 0.316i)5-s − 1.30·7-s + 0.353·8-s + (−0.387 − 0.223i)10-s − 1.89i·11-s + (−0.603 − 0.348i)13-s + (0.460 + 0.798i)14-s + (−0.125 − 0.216i)16-s + (1.02 − 0.591i)17-s + (0.917 − 0.397i)19-s + 0.316i·20-s + (−1.16 + 0.670i)22-s + (−1.02 − 0.589i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.600 + 0.799i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.600 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.386440 - 0.773914i\)
\(L(\frac12)\) \(\approx\) \(0.386440 - 0.773914i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (-4 + 1.73i)T \)
good5 \( 1 + (-1.22 + 0.707i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.44T + 7T^{2} \)
11 \( 1 + 6.29iT - 11T^{2} \)
13 \( 1 + (2.17 + 1.25i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.22 + 2.43i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.89 + 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.22 - 2.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.43iT - 31T^{2} \)
37 \( 1 - 5.97iT - 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.94 - 5.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.22 + 2.43i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.44 + 4.24i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.77 - 8.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.724 - 1.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.84 - 3.37i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.94 - 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.17 + 4.71i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.97iT - 83T^{2} \)
89 \( 1 + (-6.12 + 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.65 - 0.953i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.25076303472704195011451621708, −9.942865832322809626763763109105, −9.675514627731449595453860925708, −8.610487849298834509276573170626, −7.57372737806915186569548808838, −6.19045815224334056617204940269, −5.38613130679508868604721969862, −3.58987002371699899314197121743, −2.75871834402987645767145305650, −0.67380810995184899578608453205, 2.03417745204714624361117094823, 3.70359059311434464693172651942, 5.17520897728168386702038101616, 6.23212832585137339153002132835, 7.05981310357090466873499900579, 7.84634785043436200128682521670, 9.484664049480102927572136944692, 9.745487736972774836504317323370, 10.42117984027371807302126923724, 12.21909603951458846473096369530

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