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Properties

Label 2-342-171.113-c1-0-3
Degree $2$
Conductor $342$
Sign $0.223 - 0.974i$
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 + (−0.607 + 1.62i)3-s + (−0.499 + 0.866i)4-s + (3.41 + 1.97i)5-s + (1.70 − 0.284i)6-s + (1.07 + 1.85i)7-s + 0.999·8-s + (−2.26 − 1.97i)9-s − 3.94i·10-s + (−4.44 + 2.56i)11-s + (−1.10 − 1.33i)12-s + (−1.12 − 0.648i)13-s + (1.07 − 1.85i)14-s + (−5.26 + 4.33i)15-s + (−0.5 − 0.866i)16-s − 5.77i·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.350 + 0.936i)3-s + (−0.249 + 0.433i)4-s + (1.52 + 0.881i)5-s + (0.697 − 0.116i)6-s + (0.405 + 0.702i)7-s + 0.353·8-s + (−0.753 − 0.656i)9-s − 1.24i·10-s + (−1.33 + 0.772i)11-s + (−0.317 − 0.385i)12-s + (−0.311 − 0.179i)13-s + (0.286 − 0.496i)14-s + (−1.36 + 1.12i)15-s + (−0.125 − 0.216i)16-s − 1.40i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.223 - 0.974i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.223 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.880297 + 0.701590i\)
\(L(\frac12)\) \(\approx\) \(0.880297 + 0.701590i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.607 - 1.62i)T \)
19 \( 1 + (-1.35 - 4.14i)T \)
good5 \( 1 + (-3.41 - 1.97i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.07 - 1.85i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.44 - 2.56i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.12 + 0.648i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.77iT - 17T^{2} \)
23 \( 1 + (-1.34 - 0.778i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.09 + 1.90i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.12 - 1.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 12.0iT - 37T^{2} \)
41 \( 1 + (-2.61 + 4.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.52 + 4.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.45 + 1.41i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.42T + 53T^{2} \)
59 \( 1 + (-5.24 + 9.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0868 - 0.150i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.4 - 6.02i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.02T + 71T^{2} \)
73 \( 1 - 8.12T + 73T^{2} \)
79 \( 1 + (-0.783 + 0.452i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.1 - 6.45i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (-6.43 + 3.71i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.46774441712619543384698395435, −10.56183031393213724430981628594, −9.873994573803616555694123101158, −9.546997260180869420008617347479, −8.223353940356796999301206721798, −6.85923609629036096175198436933, −5.50209057628116300535819859654, −5.00068801725248516654777017548, −3.04887574915051374175535302191, −2.23327383231631241885711692119, 0.948889214844471972669268260000, 2.28373998739429758446643060946, 4.84583926745791724016790785737, 5.62442971935901367947063935382, 6.37750297385339455314538896431, 7.59016751438579412372228603523, 8.364694288551336008767678192127, 9.288044428512358893091279433846, 10.45183615234623511455960856083, 11.04902272082821105540474359724

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