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Properties

Label 2-363-33.5-c2-0-2
Degree $2$
Conductor $363$
Sign $-0.964 - 0.263i$
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  + (2.51 − 0.817i)2-s + (−0.853 + 2.87i)3-s + (2.42 − 1.76i)4-s + (−2.68 − 0.874i)5-s + (0.203 + 7.93i)6-s + (−6.05 + 4.39i)7-s + (−1.55 + 2.14i)8-s + (−7.54 − 4.90i)9-s − 7.48·10-s + (3.00 + 8.48i)12-s + (−6.93 − 21.3i)13-s + (−11.6 + 16.0i)14-s + (4.80 − 6.99i)15-s + (−5.87 + 18.0i)16-s + (−20.1 − 6.54i)17-s + (−22.9 − 6.18i)18-s + ⋯
L(s)  = 1  + (1.25 − 0.408i)2-s + (−0.284 + 0.958i)3-s + (0.606 − 0.440i)4-s + (−0.537 − 0.174i)5-s + (0.0339 + 1.32i)6-s + (−0.864 + 0.628i)7-s + (−0.194 + 0.267i)8-s + (−0.838 − 0.545i)9-s − 0.748·10-s + (0.250 + 0.707i)12-s + (−0.533 − 1.64i)13-s + (−0.831 + 1.14i)14-s + (0.320 − 0.466i)15-s + (−0.366 + 1.12i)16-s + (−1.18 − 0.384i)17-s + (−1.27 − 0.343i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.964 - 0.263i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ -0.964 - 0.263i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0924442 + 0.689665i\)
\(L(\frac12)\) \(\approx\) \(0.0924442 + 0.689665i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.853 - 2.87i)T \)
11 \( 1 \)
good2 \( 1 + (-2.51 + 0.817i)T + (3.23 - 2.35i)T^{2} \)
5 \( 1 + (2.68 + 0.874i)T + (20.2 + 14.6i)T^{2} \)
7 \( 1 + (6.05 - 4.39i)T + (15.1 - 46.6i)T^{2} \)
13 \( 1 + (6.93 + 21.3i)T + (-136. + 99.3i)T^{2} \)
17 \( 1 + (20.1 + 6.54i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (-12.1 - 8.79i)T + (111. + 343. i)T^{2} \)
23 \( 1 - 31.1iT - 529T^{2} \)
29 \( 1 + (-12.4 - 17.1i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (-9.27 - 28.5i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (-8.09 + 5.87i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (24.8 - 34.2i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 - 14.9T + 1.84e3T^{2} \)
47 \( 1 + (21.6 - 29.7i)T + (-682. - 2.10e3i)T^{2} \)
53 \( 1 + (40.3 - 13.1i)T + (2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-19.9 - 27.4i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-30.0 + 92.5i)T + (-3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 42T + 4.48e3T^{2} \)
71 \( 1 + (61.8 + 20.1i)T + (4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-60.5 + 43.9i)T + (1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (6.93 + 21.3i)T + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (20.1 + 6.54i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 - 62.2iT - 7.92e3T^{2} \)
97 \( 1 + (-22.8 - 70.3i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−11.82342048342734593325658715650, −10.97511758011118887210083185821, −9.963235820506454349509447547835, −9.070008607444822993919407148227, −7.969976785086212847085444400135, −6.31734176100077716669742543756, −5.42626745774935893373423584092, −4.70573406272694777679163984301, −3.46660567352332143605623499328, −2.88467536300897889755684168631, 0.20079447340799948611478264817, 2.45444758731655070931139029568, 3.88446504708916299234736015096, 4.72929125965433413546849894456, 6.16683899637883346140696086092, 6.77301746390763347786072986886, 7.33800139401629535890611146783, 8.775051043984409169294465996214, 9.936282927453056847058547139053, 11.35547056650955396211352313477

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