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Properties

Label 2-363-33.20-c2-0-29
Degree $2$
Conductor $363$
Sign $0.894 + 0.447i$
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  + (−3.41 − 1.10i)2-s + (2.93 − 0.630i)3-s + (7.17 + 5.21i)4-s + (1.98 − 0.644i)5-s + (−10.7 − 1.10i)6-s + (7.17 + 5.20i)7-s + (−10.2 − 14.1i)8-s + (8.20 − 3.69i)9-s − 7.48·10-s + (24.3 + 10.7i)12-s + (0.508 − 1.56i)13-s + (−18.6 − 25.7i)14-s + (5.41 − 3.14i)15-s + (8.39 + 25.8i)16-s + (11.8 − 3.83i)17-s + (−32.0 + 3.51i)18-s + ⋯
L(s)  = 1  + (−1.70 − 0.554i)2-s + (0.977 − 0.210i)3-s + (1.79 + 1.30i)4-s + (0.396 − 0.128i)5-s + (−1.78 − 0.183i)6-s + (1.02 + 0.744i)7-s + (−1.28 − 1.76i)8-s + (0.911 − 0.410i)9-s − 0.748·10-s + (2.02 + 0.897i)12-s + (0.0390 − 0.120i)13-s + (−1.33 − 1.83i)14-s + (0.360 − 0.209i)15-s + (0.524 + 1.61i)16-s + (0.695 − 0.225i)17-s + (−1.78 + 0.195i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.894 + 0.447i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.30450 - 0.308366i\)
\(L(\frac12)\) \(\approx\) \(1.30450 - 0.308366i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.93 + 0.630i)T \)
11 \( 1 \)
good2 \( 1 + (3.41 + 1.10i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (-1.98 + 0.644i)T + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (-7.17 - 5.20i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (-0.508 + 1.56i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (-11.8 + 3.83i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-8.54 + 6.21i)T + (111. - 343. i)T^{2} \)
23 \( 1 - 20.3iT - 529T^{2} \)
29 \( 1 + (6.82 - 9.39i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (7.22 - 22.2i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (5.86 + 4.25i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-22.8 - 31.4i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 15.8T + 1.84e3T^{2} \)
47 \( 1 + (26.6 + 36.6i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (-38.3 - 12.4i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-66.5 + 91.5i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (23.9 + 73.7i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 - 62.9T + 4.48e3T^{2} \)
71 \( 1 + (-9.71 + 3.15i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (60.3 + 43.8i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (26.6 - 82.0i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-11.2 + 3.64i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 74.5iT - 7.92e3T^{2} \)
97 \( 1 + (23.8 - 73.3i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−10.99572747426132054494858406761, −9.789906454422928759545882987977, −9.361096839334357134667149528169, −8.440059859323464514884493846933, −7.87542238879853568109521769311, −6.99175176381507686297718019924, −5.33051774106902039039080426777, −3.35979864893312394659853266186, −2.17323962917187874092954162784, −1.31926158916856479645758259274, 1.18772127531013355395984510220, 2.31867445843659785534572971597, 4.17207723704251599225221129633, 5.78752020729348632399787903271, 7.11423220851433892109042577180, 7.77698207840834899595655576238, 8.410716760908029135794965920774, 9.338064827961674463142275228241, 10.16399255734188451091190423962, 10.65047551970095048698380063629

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