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Properties

Label 2-342-171.65-c1-0-7
Degree $2$
Conductor $342$
Sign $0.993 - 0.112i$
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 + (1.29 − 1.15i)3-s + (−0.499 + 0.866i)4-s + 3.18i·5-s + (−1.64 − 0.540i)6-s + (−1.55 + 2.69i)7-s + 0.999·8-s + (0.332 − 2.98i)9-s + (2.76 − 1.59i)10-s + (1.60 + 0.925i)11-s + (0.354 + 1.69i)12-s + (4.57 + 2.64i)13-s + 3.10·14-s + (3.68 + 4.11i)15-s + (−0.5 − 0.866i)16-s + (−2.17 − 1.25i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.745 − 0.666i)3-s + (−0.249 + 0.433i)4-s + 1.42i·5-s + (−0.671 − 0.220i)6-s + (−0.587 + 1.01i)7-s + 0.353·8-s + (0.110 − 0.993i)9-s + (0.873 − 0.504i)10-s + (0.483 + 0.279i)11-s + (0.102 + 0.489i)12-s + (1.27 + 0.733i)13-s + 0.830·14-s + (0.950 + 1.06i)15-s + (−0.125 − 0.216i)16-s + (−0.528 − 0.305i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.993 - 0.112i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.993 - 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34026 + 0.0757066i\)
\(L(\frac12)\) \(\approx\) \(1.34026 + 0.0757066i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 + 1.15i)T \)
19 \( 1 + (-3.19 - 2.96i)T \)
good5 \( 1 - 3.18iT - 5T^{2} \)
7 \( 1 + (1.55 - 2.69i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.60 - 0.925i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.57 - 2.64i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.17 + 1.25i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.04 + 1.17i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.87T + 29T^{2} \)
31 \( 1 + (-7.64 + 4.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.0729iT - 37T^{2} \)
41 \( 1 + 9.99T + 41T^{2} \)
43 \( 1 + (-1.65 - 2.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12.2iT - 47T^{2} \)
53 \( 1 + (-6.17 - 10.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.48T + 59T^{2} \)
61 \( 1 + 5.54T + 61T^{2} \)
67 \( 1 + (5.91 + 3.41i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.11 + 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.73 + 8.20i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.8 + 6.26i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.54 - 2.04i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.70 + 6.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.8 - 7.97i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.77211637042743853944620117793, −10.60097583915182383966295900923, −9.570177047315838471189314317043, −8.909844225557343714821184119870, −7.87075126848813410273168306561, −6.73168753434916694993378382880, −6.16049302049459396667912420898, −3.82454946391919730110167832274, −2.95190770531098505873203613492, −1.92290761033916631459675689342, 1.09928427495444995052218161684, 3.51074489598468971663412799543, 4.44625132954081054281206287225, 5.51848103112512945545935493775, 6.82761158605072268014905602827, 8.116714093910675374717304420697, 8.625125324730753344838002571711, 9.463518266589696283123051855267, 10.24787181737770791589986545808, 11.24795585449221311514747584504

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