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Properties

Label 2-341-11.4-c1-0-3
Degree $2$
Conductor $341$
Sign $0.0301 - 0.999i$
Analytic cond. $2.72289$
Root an. cond. $1.65012$
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  + (−0.960 − 0.697i)2-s + (−0.349 + 1.07i)3-s + (−0.182 − 0.562i)4-s + (−1.04 + 0.762i)5-s + (1.08 − 0.789i)6-s + (−1.13 − 3.49i)7-s + (−0.950 + 2.92i)8-s + (1.38 + 1.00i)9-s + 1.53·10-s + (−3.20 + 0.853i)11-s + 0.670·12-s + (4.94 + 3.59i)13-s + (−1.34 + 4.14i)14-s + (−0.454 − 1.39i)15-s + (1.99 − 1.44i)16-s + (−1.97 + 1.43i)17-s + ⋯
L(s)  = 1  + (−0.678 − 0.493i)2-s + (−0.202 + 0.621i)3-s + (−0.0914 − 0.281i)4-s + (−0.469 + 0.341i)5-s + (0.443 − 0.322i)6-s + (−0.428 − 1.31i)7-s + (−0.336 + 1.03i)8-s + (0.463 + 0.336i)9-s + 0.486·10-s + (−0.966 + 0.257i)11-s + 0.193·12-s + (1.37 + 0.997i)13-s + (−0.359 + 1.10i)14-s + (−0.117 − 0.360i)15-s + (0.498 − 0.362i)16-s + (−0.479 + 0.348i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0301 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0301 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(341\)    =    \(11 \cdot 31\)
Sign: $0.0301 - 0.999i$
Analytic conductor: \(2.72289\)
Root analytic conductor: \(1.65012\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{341} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 341,\ (\ :1/2),\ 0.0301 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.316708 + 0.307294i\)
\(L(\frac12)\) \(\approx\) \(0.316708 + 0.307294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (3.20 - 0.853i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (0.960 + 0.697i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.349 - 1.07i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.04 - 0.762i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.13 + 3.49i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.94 - 3.59i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.97 - 1.43i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.06 - 3.26i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.88T + 23T^{2} \)
29 \( 1 + (-3.14 - 9.66i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (0.107 + 0.331i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.24 + 6.89i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.12T + 43T^{2} \)
47 \( 1 + (1.69 - 5.20i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.58 + 2.60i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.176 - 0.542i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.80 - 4.21i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 5.96T + 67T^{2} \)
71 \( 1 + (-9.56 + 6.94i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.18 + 9.80i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.31 - 4.58i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.4 - 8.35i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 0.0753T + 89T^{2} \)
97 \( 1 + (12.2 + 8.89i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.08155841972264415052787648097, −10.75216696818863702594987307976, −10.18923685727460971859208364902, −9.256603015118485451740850610407, −8.124940894051581754518592709257, −7.14997988513885092258526155035, −5.90619751450810780386432728877, −4.48885661953961819204238497566, −3.65642837848483623366055449966, −1.66153332322902507968771582908, 0.40060934808717325589053932219, 2.70466105527783520710227662733, 4.14120658380775220723193718295, 5.86285140420248378488723570370, 6.45907468379782570680744060496, 7.87719213347972591356740534009, 8.229080619549784075313468037570, 9.174056927655346882213222096624, 10.14401070080152932705811738952, 11.51624566475804190083164070659

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