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Properties

Label 2-342-171.110-c1-0-5
Degree $2$
Conductor $342$
Sign $-0.198 - 0.980i$
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  + (−0.766 + 0.642i)2-s + (1.68 + 0.386i)3-s + (0.173 − 0.984i)4-s + (−0.944 + 2.59i)5-s + (−1.54 + 0.789i)6-s + (−1.67 − 2.90i)7-s + (0.500 + 0.866i)8-s + (2.70 + 1.30i)9-s + (−0.944 − 2.59i)10-s + 6.01i·11-s + (0.673 − 1.59i)12-s + (1.37 + 3.77i)13-s + (3.15 + 1.14i)14-s + (−2.59 + 4.01i)15-s + (−0.939 − 0.342i)16-s + (1.08 − 2.99i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (0.974 + 0.223i)3-s + (0.0868 − 0.492i)4-s + (−0.422 + 1.16i)5-s + (−0.629 + 0.322i)6-s + (−0.634 − 1.09i)7-s + (0.176 + 0.306i)8-s + (0.900 + 0.434i)9-s + (−0.298 − 0.820i)10-s + 1.81i·11-s + (0.194 − 0.460i)12-s + (0.380 + 1.04i)13-s + (0.843 + 0.306i)14-s + (−0.670 + 1.03i)15-s + (−0.234 − 0.0855i)16-s + (0.264 − 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.765072 + 0.935199i\)
\(L(\frac12)\) \(\approx\) \(0.765072 + 0.935199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (-1.68 - 0.386i)T \)
19 \( 1 + (1.90 - 3.92i)T \)
good5 \( 1 + (0.944 - 2.59i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (1.67 + 2.90i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 6.01iT - 11T^{2} \)
13 \( 1 + (-1.37 - 3.77i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.08 + 2.99i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-1.67 - 0.294i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.20 + 6.85i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 - 1.49iT - 31T^{2} \)
37 \( 1 + 9.47iT - 37T^{2} \)
41 \( 1 + (4.60 - 3.86i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.275 + 1.56i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (1.70 + 0.301i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-1.37 - 1.15i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-0.982 - 5.56i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-12.3 + 4.50i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-6.43 + 7.66i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (6.75 - 5.66i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.28 + 7.30i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-2.75 + 7.55i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-1.22 + 0.707i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.88 + 16.3i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-6.43 - 7.67i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.56611573976521689677631622671, −10.34179821054382374333249895772, −10.00944214882980419735548955596, −9.108662125284343160311026364714, −7.72233642021765678468409092506, −7.19304333167185453500549691321, −6.59560993566431431913884333261, −4.48622121057994813937793381183, −3.60025060930191717338600125868, −2.09483306004347015504427946569, 0.962782990405519435141164266860, 2.80548495569906286327833836623, 3.61647281438857763875712605727, 5.28540744826994682113312566874, 6.53578185585563993704767111475, 8.158501575838966110516088579240, 8.571965585308010738243355602795, 8.985958337341515007322369933366, 10.17734145299359773544846259045, 11.30203717940656355094162110309

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