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Properties

Label 2-348-29.17-c2-0-9
Degree $2$
Conductor $348$
Sign $-0.999 - 0.0147i$
Analytic cond. $9.48231$
Root an. cond. $3.07933$
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 − 1.22i)3-s − 6.22i·5-s − 8.38·7-s − 2.99i·9-s + (−11.2 + 11.2i)11-s − 12.5i·13-s + (−7.62 − 7.62i)15-s + (−2.97 + 2.97i)17-s + (−19.0 + 19.0i)19-s + (−10.2 + 10.2i)21-s − 0.990·23-s − 13.7·25-s + (−3.67 − 3.67i)27-s + (23.7 + 16.6i)29-s + (9.11 − 9.11i)31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s − 1.24i·5-s − 1.19·7-s − 0.333i·9-s + (−1.02 + 1.02i)11-s − 0.965i·13-s + (−0.508 − 0.508i)15-s + (−0.175 + 0.175i)17-s + (−1.00 + 1.00i)19-s + (−0.489 + 0.489i)21-s − 0.0430·23-s − 0.549·25-s + (−0.136 − 0.136i)27-s + (0.819 + 0.572i)29-s + (0.294 − 0.294i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0147i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(348\)    =    \(2^{2} \cdot 3 \cdot 29\)
Sign: $-0.999 - 0.0147i$
Analytic conductor: \(9.48231\)
Root analytic conductor: \(3.07933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{348} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 348,\ (\ :1),\ -0.999 - 0.0147i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00471084 + 0.636624i\)
\(L(\frac12)\) \(\approx\) \(0.00471084 + 0.636624i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.22 + 1.22i)T \)
29 \( 1 + (-23.7 - 16.6i)T \)
good5 \( 1 + 6.22iT - 25T^{2} \)
7 \( 1 + 8.38T + 49T^{2} \)
11 \( 1 + (11.2 - 11.2i)T - 121iT^{2} \)
13 \( 1 + 12.5iT - 169T^{2} \)
17 \( 1 + (2.97 - 2.97i)T - 289iT^{2} \)
19 \( 1 + (19.0 - 19.0i)T - 361iT^{2} \)
23 \( 1 + 0.990T + 529T^{2} \)
31 \( 1 + (-9.11 + 9.11i)T - 961iT^{2} \)
37 \( 1 + (41.7 + 41.7i)T + 1.36e3iT^{2} \)
41 \( 1 + (18.0 + 18.0i)T + 1.68e3iT^{2} \)
43 \( 1 + (-50.0 + 50.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (53.3 + 53.3i)T + 2.20e3iT^{2} \)
53 \( 1 + 42.5T + 2.80e3T^{2} \)
59 \( 1 + 61.9T + 3.48e3T^{2} \)
61 \( 1 + (-36.0 + 36.0i)T - 3.72e3iT^{2} \)
67 \( 1 + 66.8iT - 4.48e3T^{2} \)
71 \( 1 - 89.1iT - 5.04e3T^{2} \)
73 \( 1 + (-16.0 - 16.0i)T + 5.32e3iT^{2} \)
79 \( 1 + (-74.3 + 74.3i)T - 6.24e3iT^{2} \)
83 \( 1 + 80.2T + 6.88e3T^{2} \)
89 \( 1 + (-3.67 + 3.67i)T - 7.92e3iT^{2} \)
97 \( 1 + (101. + 101. i)T + 9.40e3iT^{2} \)
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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

−10.51847734571255999332134970955, −9.892905025183669738778452461620, −8.830164574665168914643776972683, −8.119381292108108954986961312526, −7.07752519579279975785478539611, −5.88603437049256263271188126426, −4.82970722178775892690435572180, −3.48654855033394986883634451435, −2.05301328010160642264100341522, −0.25626066050974791649757745422, 2.66398904711372127556911288060, 3.21290197927113032046764689149, 4.63141188306545186001729695429, 6.26742483785636994243974760328, 6.74523863473182479797848506318, 8.039671318709402531467895314558, 9.068641745147394055167858063031, 9.994525429626192835224261677641, 10.73492668578197712773033863323, 11.43166335871463157866189814845

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