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Properties

Label 2-363-33.5-c2-0-25
Degree $2$
Conductor $363$
Sign $0.701 + 0.712i$
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  + (−0.951 + 0.309i)2-s + (0.303 + 2.98i)3-s + (−2.42 + 1.76i)4-s + (−6.29 − 2.04i)5-s + (−1.21 − 2.74i)6-s + (−7.16 + 5.20i)7-s + (4.11 − 5.66i)8-s + (−8.81 + 1.81i)9-s + 6.61·10-s + (−6 − 6.70i)12-s + (4.09 + 12.5i)13-s + (5.20 − 7.16i)14-s + (4.19 − 19.4i)15-s + (1.54 − 4.75i)16-s + (−5.70 − 1.85i)17-s + (7.82 − 4.44i)18-s + ⋯
L(s)  = 1  + (−0.475 + 0.154i)2-s + (0.101 + 0.994i)3-s + (−0.606 + 0.440i)4-s + (−1.25 − 0.409i)5-s + (−0.201 − 0.457i)6-s + (−1.02 + 0.743i)7-s + (0.514 − 0.707i)8-s + (−0.979 + 0.201i)9-s + 0.661·10-s + (−0.5 − 0.559i)12-s + (0.314 + 0.968i)13-s + (0.371 − 0.511i)14-s + (0.279 − 1.29i)15-s + (0.0965 − 0.297i)16-s + (−0.335 − 0.109i)17-s + (0.434 − 0.247i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.701 + 0.712i$
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: \((2,\ 363,\ (\ :1),\ 0.701 + 0.712i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.131528 - 0.0551078i\)
\(L(\frac12)\) \(\approx\) \(0.131528 - 0.0551078i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.303 - 2.98i)T \)
11 \( 1 \)
good2 \( 1 + (0.951 - 0.309i)T + (3.23 - 2.35i)T^{2} \)
5 \( 1 + (6.29 + 2.04i)T + (20.2 + 14.6i)T^{2} \)
7 \( 1 + (7.16 - 5.20i)T + (15.1 - 46.6i)T^{2} \)
13 \( 1 + (-4.09 - 12.5i)T + (-136. + 99.3i)T^{2} \)
17 \( 1 + (5.70 + 1.85i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (-7.38 - 5.36i)T + (111. + 343. i)T^{2} \)
23 \( 1 + 17.5iT - 529T^{2} \)
29 \( 1 + (-15.5 - 21.3i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (1.91 + 5.89i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (-15.8 + 11.5i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (3.28 - 4.52i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 - 26.2T + 1.84e3T^{2} \)
47 \( 1 + (10.4 - 14.3i)T + (-682. - 2.10e3i)T^{2} \)
53 \( 1 + (21.1 - 6.88i)T + (2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (12.8 + 17.7i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-28.8 + 88.8i)T + (-3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 76.7T + 4.48e3T^{2} \)
71 \( 1 + (62.5 + 20.3i)T + (4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-12.1 + 8.82i)T + (1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (36.4 + 112. i)T + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-105. - 34.3i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + 97.6iT - 7.92e3T^{2} \)
97 \( 1 + (37.4 + 115. i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−11.04553622992431565359707974681, −9.865014720860241195947448542227, −9.077705605707178062104726160064, −8.629605515443674189665850472213, −7.64506171481640690521754169776, −6.32800346351236212798519409359, −4.80883497911254183413809336744, −4.03089567377829656355761640929, −3.10852733987554667679333683404, −0.10417238727931920115215521553, 0.927271556580272657579883864751, 2.98820251425583955047059321844, 4.06595769827183158643276771899, 5.66764780197134799382498038273, 6.82646461913787917109256637101, 7.67405416805943116021143198298, 8.346731624360807812551042567720, 9.462597114860066673948874601135, 10.47186368310246446169897305947, 11.24685751172747059898088030540

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