Properties

Label 2-19e2-19.10-c2-0-39
Degree $2$
Conductor $361$
Sign $0.999 - 0.0158i$
Analytic cond. $9.83653$
Root an. cond. $3.13632$
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  + (3.55 − 0.626i)2-s + (2.31 + 2.76i)3-s + (8.45 − 3.07i)4-s + (−3.75 − 1.36i)5-s + (9.95 + 8.35i)6-s + (2.5 − 4.33i)7-s + (15.6 − 9.01i)8-s + (−0.694 + 3.93i)9-s + (−14.2 − 2.50i)10-s + (5 + 8.66i)11-s + (28.1 + 16.2i)12-s + (2.31 − 2.76i)13-s + (6.16 − 16.9i)14-s + (−4.93 − 13.5i)15-s + (22.2 − 18.6i)16-s + (2.60 + 14.7i)17-s + ⋯
L(s)  = 1  + (1.77 − 0.313i)2-s + (0.772 + 0.920i)3-s + (2.11 − 0.769i)4-s + (−0.751 − 0.273i)5-s + (1.65 + 1.39i)6-s + (0.357 − 0.618i)7-s + (1.95 − 1.12i)8-s + (−0.0771 + 0.437i)9-s + (−1.42 − 0.250i)10-s + (0.454 + 0.787i)11-s + (2.34 + 1.35i)12-s + (0.178 − 0.212i)13-s + (0.440 − 1.21i)14-s + (−0.328 − 0.903i)15-s + (1.38 − 1.16i)16-s + (0.153 + 0.868i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0158i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.999 - 0.0158i$
Analytic conductor: \(9.83653\)
Root analytic conductor: \(3.13632\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 361,\ (\ :1),\ 0.999 - 0.0158i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.44982 + 0.0430698i\)
\(L(\frac12)\) \(\approx\) \(5.44982 + 0.0430698i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-3.55 + 0.626i)T + (3.75 - 1.36i)T^{2} \)
3 \( 1 + (-2.31 - 2.76i)T + (-1.56 + 8.86i)T^{2} \)
5 \( 1 + (3.75 + 1.36i)T + (19.1 + 16.0i)T^{2} \)
7 \( 1 + (-2.5 + 4.33i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-5 - 8.66i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-2.31 + 2.76i)T + (-29.3 - 166. i)T^{2} \)
17 \( 1 + (-2.60 - 14.7i)T + (-271. + 98.8i)T^{2} \)
23 \( 1 + (32.8 - 11.9i)T + (405. - 340. i)T^{2} \)
29 \( 1 + (17.7 + 3.13i)T + (790. + 287. i)T^{2} \)
31 \( 1 + (31.2 + 18.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 21.6iT - 1.36e3T^{2} \)
41 \( 1 + (23.1 + 27.6i)T + (-291. + 1.65e3i)T^{2} \)
43 \( 1 + (-18.7 - 6.84i)T + (1.41e3 + 1.18e3i)T^{2} \)
47 \( 1 + (-1.73 + 9.84i)T + (-2.07e3 - 755. i)T^{2} \)
53 \( 1 + (25.8 + 71.1i)T + (-2.15e3 + 1.80e3i)T^{2} \)
59 \( 1 + (-17.7 + 3.13i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (-37.5 + 13.6i)T + (2.85e3 - 2.39e3i)T^{2} \)
67 \( 1 + (39.0 + 6.88i)T + (4.21e3 + 1.53e3i)T^{2} \)
71 \( 1 + (36.9 - 101. i)T + (-3.86e3 - 3.24e3i)T^{2} \)
73 \( 1 + (-80.4 + 67.4i)T + (925. - 5.24e3i)T^{2} \)
79 \( 1 + (-23.1 - 27.6i)T + (-1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (-20 + 34.6i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-1.37e3 - 7.80e3i)T^{2} \)
97 \( 1 + (-120. + 21.2i)T + (8.84e3 - 3.21e3i)T^{2} \)
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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

−11.48448478620711830356213342342, −10.51045439834311794588693523703, −9.692356525238244028685708457983, −8.295162159429798387313851862752, −7.31962288684979629638721043562, −6.02704605101775934761275281363, −4.77577621302871347733451406828, −3.88329696443821777750615696154, −3.70126719452942001084058263422, −1.97166512336513912544936655773, 2.00425444569630538487625309990, 3.10371183725012278969069858538, 4.01743146315334454647233521149, 5.33491110555558247255291680490, 6.33133944685503304250372500035, 7.30518114962564762860197935746, 7.939800414273355335190259887991, 8.985206954507635051951743688776, 10.91521705215267139837878897894, 11.73470940324752283284400161828

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