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Properties

Label 2-363-33.8-c1-0-12
Degree $2$
Conductor $363$
Sign $0.530 - 0.847i$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
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  + (0.809 − 0.587i)2-s + (1.65 + 0.514i)3-s + (−0.309 + 0.951i)4-s + (−1.66 + 2.28i)5-s + (1.64 − 0.556i)6-s + (−1.34 − 0.437i)7-s + (0.927 + 2.85i)8-s + (2.47 + 1.70i)9-s + 2.82i·10-s + (−0.999 + 1.41i)12-s + (−2.49 − 3.43i)13-s + (−1.34 + 0.437i)14-s + (−3.92 + 2.93i)15-s + (0.809 + 0.587i)16-s + (4.85 + 3.52i)17-s + (2.99 − 0.0770i)18-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.954 + 0.296i)3-s + (−0.154 + 0.475i)4-s + (−0.743 + 1.02i)5-s + (0.669 − 0.227i)6-s + (−0.508 − 0.165i)7-s + (0.327 + 1.00i)8-s + (0.823 + 0.566i)9-s + 0.894i·10-s + (−0.288 + 0.408i)12-s + (−0.691 − 0.951i)13-s + (−0.359 + 0.116i)14-s + (−1.01 + 0.756i)15-s + (0.202 + 0.146i)16-s + (1.17 + 0.855i)17-s + (0.706 − 0.0181i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.530 - 0.847i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.530 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73723 + 0.962002i\)
\(L(\frac12)\) \(\approx\) \(1.73723 + 0.962002i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 - 0.514i)T \)
11 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (1.66 - 2.28i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.34 + 0.437i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.49 + 3.43i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.85 - 3.52i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.03 + 1.31i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-0.618 + 1.90i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.61 + 1.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.47 + 7.60i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.85 + 5.70i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 + (-2.68 + 0.874i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.32 - 4.57i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-10.7 - 3.49i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.81 - 8.00i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (1.66 - 2.28i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.34 + 0.437i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.49 + 3.43i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.9 + 9.40i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-1.61 + 1.17i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.71225329415024812608518134465, −10.59199024303321428412952517140, −9.978527295176168162605108132782, −8.693247557969835691089685168807, −7.64397670479336097202130499956, −7.31665161863667453914891729858, −5.47515682830857144369603589354, −4.04369940524527576437865870621, −3.35559728162297021372359373503, −2.62065961282951665943360494335, 1.17040102642140036092887412283, 3.17872826537089059940287612674, 4.36064224413887219174121444568, 5.17471733294921790138504361036, 6.58037278649889231894706433434, 7.47528771077007218341320583062, 8.406746022705463016744436354232, 9.599857552294104071189633205320, 9.766185125658401753727480702019, 11.70471053207131668721890179431

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