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Properties

Label 2-342-171.151-c2-0-34
Degree $2$
Conductor $342$
Sign $-0.190 + 0.981i$
Analytic cond. $9.31882$
Root an. cond. $3.05267$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (2.61 − 1.46i)3-s + (0.999 + 1.73i)4-s + (0.350 + 0.607i)5-s + (−4.24 − 0.0540i)6-s + (5.86 − 10.1i)7-s − 2.82i·8-s + (4.69 − 7.67i)9-s − 0.992i·10-s + (0.375 − 0.649i)11-s + (5.15 + 3.06i)12-s + (−11.4 + 6.61i)13-s + (−14.3 + 8.29i)14-s + (1.80 + 1.07i)15-s + (−2.00 + 3.46i)16-s − 8.97·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.872 − 0.488i)3-s + (0.249 + 0.433i)4-s + (0.0701 + 0.121i)5-s + (−0.707 − 0.00900i)6-s + (0.838 − 1.45i)7-s − 0.353i·8-s + (0.521 − 0.853i)9-s − 0.0992i·10-s + (0.0340 − 0.0590i)11-s + (0.429 + 0.255i)12-s + (−0.881 + 0.508i)13-s + (−1.02 + 0.592i)14-s + (0.120 + 0.0716i)15-s + (−0.125 + 0.216i)16-s − 0.528·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.190 + 0.981i$
Analytic conductor: \(9.31882\)
Root analytic conductor: \(3.05267\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1),\ -0.190 + 0.981i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.10438 - 1.33939i\)
\(L(\frac12)\) \(\approx\) \(1.10438 - 1.33939i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (-2.61 + 1.46i)T \)
19 \( 1 + (-16.3 + 9.63i)T \)
good5 \( 1 + (-0.350 - 0.607i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-5.86 + 10.1i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-0.375 + 0.649i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (11.4 - 6.61i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 8.97T + 289T^{2} \)
23 \( 1 + (-9.47 - 16.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (1.14 + 0.663i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (5.23 - 3.02i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 42.1iT - 1.36e3T^{2} \)
41 \( 1 + (43.2 - 24.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (1.35 - 2.34i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-5.09 + 8.82i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 46.2iT - 2.80e3T^{2} \)
59 \( 1 + (-54.7 + 31.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-26.1 + 45.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (34.9 - 20.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 6.83iT - 5.04e3T^{2} \)
73 \( 1 - 54.2T + 5.32e3T^{2} \)
79 \( 1 + (-78.2 - 45.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (40.4 - 70.0i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 1.18iT - 7.92e3T^{2} \)
97 \( 1 + (-25.8 - 14.9i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.98692648958422647593567408842, −10.01559743076895481363824649188, −9.211436109632278172804839087707, −8.174613394038891479067603078292, −7.34266207866704911785740984716, −6.83355071732208903728589613690, −4.74156564278393271916926733805, −3.60690150272966145905801744438, −2.21719369232414113776636900480, −0.934843640961993532032686722988, 1.86382861620150116224649774476, 2.96247079083042952197979425929, 4.82151654772119337113780455193, 5.50236973998182963684780575853, 7.08915919076085927836644573255, 8.115279183630523986067350184251, 8.724835806920495253348362316388, 9.476604367834701055085844116793, 10.36098549127434198382324736369, 11.44915242375834543721262079299

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