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Properties

Label 2-363-33.26-c2-0-58
Degree $2$
Conductor $363$
Sign $-0.995 + 0.0913i$
Analytic cond. $9.89103$
Root an. cond. $3.14500$
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  + (1.94 − 2.68i)2-s + (0.927 − 2.85i)3-s + (−2.16 − 6.65i)4-s + (3.89 + 5.36i)5-s + (−5.84 − 8.04i)6-s + (−2.47 − 7.60i)7-s + (−9.46 − 3.07i)8-s + (−7.28 − 5.29i)9-s + 22·10-s − 21.0·12-s + (−3.23 − 2.35i)13-s + (−25.2 − 8.19i)14-s + (18.9 − 6.14i)15-s + (−4.04 + 2.93i)16-s + (−7.79 − 10.7i)17-s + (−28.3 + 9.22i)18-s + ⋯
L(s)  = 1  + (0.974 − 1.34i)2-s + (0.309 − 0.951i)3-s + (−0.540 − 1.66i)4-s + (0.779 + 1.07i)5-s + (−0.974 − 1.34i)6-s + (−0.353 − 1.08i)7-s + (−1.18 − 0.384i)8-s + (−0.809 − 0.587i)9-s + 2.20·10-s − 1.75·12-s + (−0.248 − 0.180i)13-s + (−1.80 − 0.585i)14-s + (1.26 − 0.409i)15-s + (−0.252 + 0.183i)16-s + (−0.458 − 0.631i)17-s + (−1.57 + 0.512i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0913i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.995 + 0.0913i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ -0.995 + 0.0913i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.141808 - 3.09682i\)
\(L(\frac12)\) \(\approx\) \(0.141808 - 3.09682i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.927 + 2.85i)T \)
11 \( 1 \)
good2 \( 1 + (-1.94 + 2.68i)T + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (-3.89 - 5.36i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (2.47 + 7.60i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (3.23 + 2.35i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (7.79 + 10.7i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (1.85 - 5.70i)T + (-292. - 212. i)T^{2} \)
23 \( 1 - 6.63iT - 529T^{2} \)
29 \( 1 + (-37.8 + 12.2i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-21.0 - 15.2i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-9.27 - 28.5i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (-12.6 - 4.09i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 - 42T + 1.84e3T^{2} \)
47 \( 1 + (-82.0 - 26.6i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (35.0 - 48.2i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (63.0 - 20.4i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (9.70 - 7.05i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 2T + 4.48e3T^{2} \)
71 \( 1 + (-35.0 - 48.2i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (22.8 + 70.3i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-32.3 - 23.5i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-23.3 - 32.1i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + 119. iT - 7.92e3T^{2} \)
97 \( 1 + (50.1 + 36.4i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−10.80580583669538415929062533079, −10.28922177738410651288409320956, −9.354061683405334885505482632881, −7.69406622841845970142354195536, −6.73999094045981309865208776661, −5.90848637152071194451445397243, −4.39178431927640662500296010071, −3.08704363014532725011723297926, −2.47286256906185713870686067867, −1.04910391200963713351635649266, 2.53846093966888921825394935714, 4.10979886141418239444105270536, 4.96309182690006931302688533128, 5.66607452807719524566560736091, 6.44101820869565065366513434227, 8.018399921899404995412604169145, 8.853940101272056151241176674261, 9.353212979278889004744219804001, 10.59656583858379030956853630008, 12.13079969572613691411174259569

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