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Properties

Label 2-342-9.7-c1-0-13
Degree $2$
Conductor $342$
Sign $-0.338 + 0.941i$
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 + (0.147 − 1.72i)3-s + (−0.499 − 0.866i)4-s + (−0.573 − 0.992i)5-s + (1.42 + 0.990i)6-s + (0.817 − 1.41i)7-s + 0.999·8-s + (−2.95 − 0.509i)9-s + 1.14·10-s + (−1.10 + 1.91i)11-s + (−1.56 + 0.735i)12-s + (−2.18 − 3.78i)13-s + (0.817 + 1.41i)14-s + (−1.79 + 0.842i)15-s + (−0.5 + 0.866i)16-s − 1.42·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.0852 − 0.996i)3-s + (−0.249 − 0.433i)4-s + (−0.256 − 0.443i)5-s + (0.580 + 0.404i)6-s + (0.308 − 0.535i)7-s + 0.353·8-s + (−0.985 − 0.169i)9-s + 0.362·10-s + (−0.333 + 0.576i)11-s + (−0.452 + 0.212i)12-s + (−0.605 − 1.04i)13-s + (0.218 + 0.378i)14-s + (−0.464 + 0.217i)15-s + (−0.125 + 0.216i)16-s − 0.345·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.338 + 0.941i$
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: \((2,\ 342,\ (\ :1/2),\ -0.338 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.457064 - 0.650090i\)
\(L(\frac12)\) \(\approx\) \(0.457064 - 0.650090i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.147 + 1.72i)T \)
19 \( 1 + T \)
good5 \( 1 + (0.573 + 0.992i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.817 + 1.41i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.10 - 1.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.18 + 3.78i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.42T + 17T^{2} \)
23 \( 1 + (2.16 + 3.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.700 - 1.21i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.66 + 6.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.94T + 37T^{2} \)
41 \( 1 + (-2.64 - 4.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.37 + 7.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.53 - 2.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.16T + 53T^{2} \)
59 \( 1 + (0.653 + 1.13i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.535 + 0.927i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.57 + 6.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 - 6.78T + 73T^{2} \)
79 \( 1 + (5.40 - 9.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.59 + 4.50i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.62T + 89T^{2} \)
97 \( 1 + (7.77 - 13.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.21324934157030824816690567271, −10.27593885852171624595472242725, −9.130803417497186377072975002424, −8.023872141510551434059793105325, −7.66914341217654296363715256918, −6.61961180207152625684353094177, −5.51336896395046693085100311336, −4.35477889447564127918295044970, −2.38282486184412459151644997756, −0.60357945115545643217800997962, 2.32133545515724557271107089556, 3.49393296975188857555185547584, 4.59754660020514114028847151010, 5.73558492823965381633379235030, 7.24193013538287482869539384479, 8.418149993854870943727255641457, 9.162970871967954570470430354296, 9.955740146811383135039822066309, 11.06415568868868592490212064359, 11.36630107695160997633217896905

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