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Properties

Label 2-342-171.106-c1-0-8
Degree $2$
Conductor $342$
Sign $-0.0699 + 0.997i$
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 + (−0.724 + 1.57i)3-s + (−0.499 + 0.866i)4-s − 3·5-s + (1.72 − 0.158i)6-s + (−0.724 + 1.25i)7-s + 0.999·8-s + (−1.94 − 2.28i)9-s + (1.5 + 2.59i)10-s + (2.72 − 4.71i)11-s + (−1 − 1.41i)12-s + (2 − 3.46i)13-s + 1.44·14-s + (2.17 − 4.71i)15-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.418 + 0.908i)3-s + (−0.249 + 0.433i)4-s − 1.34·5-s + (0.704 − 0.0648i)6-s + (−0.273 + 0.474i)7-s + 0.353·8-s + (−0.649 − 0.760i)9-s + (0.474 + 0.821i)10-s + (0.821 − 1.42i)11-s + (−0.288 − 0.408i)12-s + (0.554 − 0.960i)13-s + 0.387·14-s + (0.561 − 1.21i)15-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0699 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0699 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.0699 + 0.997i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.0699 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.349971 - 0.375383i\)
\(L(\frac12)\) \(\approx\) \(0.349971 - 0.375383i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.724 - 1.57i)T \)
19 \( 1 + (-1 - 4.24i)T \)
good5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + (0.724 - 1.25i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.72 + 4.71i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.44 + 4.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.89T + 29T^{2} \)
31 \( 1 + (0.724 + 1.25i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.898T + 37T^{2} \)
41 \( 1 + 1.89T + 41T^{2} \)
43 \( 1 + (5.89 + 10.2i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.55T + 47T^{2} \)
53 \( 1 + (3.94 + 6.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.550T + 59T^{2} \)
61 \( 1 + 5.89T + 61T^{2} \)
67 \( 1 + (-6.89 + 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.72 + 9.91i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.39 + 4.15i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.825 + 1.43i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2 - 3.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.25683953329631976978160939116, −10.61528272313769626683283967079, −9.446294977046855749090231033967, −8.633864814015109714865276872515, −7.87824937255457437831710729348, −6.28973841439666692584788749481, −5.17846831008434663801857825547, −3.63577691749232161008225368006, −3.39143098255594304211275737768, −0.46433179311012833485416628946, 1.46932822260268923085497819040, 3.77655761583823525315960162629, 4.85753105230035031225076443198, 6.39037164027256734620367290971, 7.15666727216791398861814276624, 7.61965664169481997969445623382, 8.754558895014385451411855374654, 9.757806225191435108547319630045, 11.20716143069474779546900997374, 11.59291631426428081647178654759

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