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Properties

Label 2-342-171.7-c1-0-18
Degree $2$
Conductor $342$
Sign $-0.0918 + 0.995i$
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  − 2-s + (1.09 − 1.34i)3-s + 4-s + (0.789 − 1.36i)5-s + (−1.09 + 1.34i)6-s + (2.31 − 4.01i)7-s − 8-s + (−0.601 − 2.93i)9-s + (−0.789 + 1.36i)10-s + (−1.83 + 3.18i)11-s + (1.09 − 1.34i)12-s − 5.57·13-s + (−2.31 + 4.01i)14-s + (−0.970 − 2.55i)15-s + 16-s + (3.65 + 6.32i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.632 − 0.774i)3-s + 0.5·4-s + (0.353 − 0.611i)5-s + (−0.447 + 0.547i)6-s + (0.875 − 1.51i)7-s − 0.353·8-s + (−0.200 − 0.979i)9-s + (−0.249 + 0.432i)10-s + (−0.554 + 0.959i)11-s + (0.316 − 0.387i)12-s − 1.54·13-s + (−0.618 + 1.07i)14-s + (−0.250 − 0.660i)15-s + 0.250·16-s + (0.886 + 1.53i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0918 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0918 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.0918 + 0.995i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.0918 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.844656 - 0.926109i\)
\(L(\frac12)\) \(\approx\) \(0.844656 - 0.926109i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-1.09 + 1.34i)T \)
19 \( 1 + (-3.55 + 2.52i)T \)
good5 \( 1 + (-0.789 + 1.36i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.31 + 4.01i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.83 - 3.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.57T + 13T^{2} \)
17 \( 1 + (-3.65 - 6.32i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + 2.10T + 23T^{2} \)
29 \( 1 + (2.66 + 4.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.587 - 1.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.16T + 37T^{2} \)
41 \( 1 + (-2.24 + 3.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 8.49T + 43T^{2} \)
47 \( 1 + (-2.87 - 4.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.69 + 2.94i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.992 + 1.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.59 + 2.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + (-4.03 - 6.98i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.45 - 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 5.98T + 79T^{2} \)
83 \( 1 + (5.85 - 10.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.00 + 5.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−11.17755461830295813676461528129, −10.05420430582345893215783175621, −9.547330993603302015835219945917, −8.183004096252365662031883157343, −7.60244513780437962627347475842, −7.03895847733118370658344165761, −5.39109142775604768230114479520, −4.07691850541685373411662560498, −2.25757392145644032994494190310, −1.09927650825934421372709717004, 2.37646439622120248518765801705, 2.97161637747161958808491275139, 5.06763702646047364743395952621, 5.68329529011908580689032590718, 7.45405317187766675733378694608, 8.107454417330134024699741875097, 9.157814359678546064702857749534, 9.719408343581446993430449811189, 10.65922616684798618754874740794, 11.58118667296291015188969326265

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