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Properties

Label 2-333-37.8-c2-0-8
Degree $2$
Conductor $333$
Sign $-0.678 - 0.734i$
Analytic cond. $9.07359$
Root an. cond. $3.01224$
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  + (2.77 + 0.742i)2-s + (3.67 + 2.11i)4-s + (−5.87 + 1.57i)5-s + (−4.19 + 7.26i)7-s + (0.486 + 0.486i)8-s − 17.4·10-s + 17.4i·11-s + (19.1 − 5.13i)13-s + (−17.0 + 17.0i)14-s + (−7.49 − 12.9i)16-s + (−6.86 + 25.6i)17-s + (−21.4 + 5.73i)19-s + (−24.9 − 6.67i)20-s + (−12.9 + 48.5i)22-s + (−9.10 − 9.10i)23-s + ⋯
L(s)  = 1  + (1.38 + 0.371i)2-s + (0.917 + 0.529i)4-s + (−1.17 + 0.314i)5-s + (−0.598 + 1.03i)7-s + (0.0607 + 0.0607i)8-s − 1.74·10-s + 1.59i·11-s + (1.47 − 0.394i)13-s + (−1.21 + 1.21i)14-s + (−0.468 − 0.811i)16-s + (−0.403 + 1.50i)17-s + (−1.12 + 0.301i)19-s + (−1.24 − 0.333i)20-s + (−0.590 + 2.20i)22-s + (−0.395 − 0.395i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333\)    =    \(3^{2} \cdot 37\)
Sign: $-0.678 - 0.734i$
Analytic conductor: \(9.07359\)
Root analytic conductor: \(3.01224\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{333} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 333,\ (\ :1),\ -0.678 - 0.734i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.822057 + 1.87855i\)
\(L(\frac12)\) \(\approx\) \(0.822057 + 1.87855i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 + (-36.9 + 0.922i)T \)
good2 \( 1 + (-2.77 - 0.742i)T + (3.46 + 2i)T^{2} \)
5 \( 1 + (5.87 - 1.57i)T + (21.6 - 12.5i)T^{2} \)
7 \( 1 + (4.19 - 7.26i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 - 17.4iT - 121T^{2} \)
13 \( 1 + (-19.1 + 5.13i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (6.86 - 25.6i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (21.4 - 5.73i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (9.10 + 9.10i)T + 529iT^{2} \)
29 \( 1 + (3.87 - 3.87i)T - 841iT^{2} \)
31 \( 1 + (-42.9 + 42.9i)T - 961iT^{2} \)
41 \( 1 + (-31.1 - 17.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-13.2 - 13.2i)T + 1.84e3iT^{2} \)
47 \( 1 + 0.00879T + 2.20e3T^{2} \)
53 \( 1 + (-4.00 - 6.94i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-7.46 + 27.8i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-10.3 - 38.4i)T + (-3.22e3 + 1.86e3i)T^{2} \)
67 \( 1 + (0.601 + 0.347i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (31.8 - 55.2i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 - 62.8iT - 5.32e3T^{2} \)
79 \( 1 + (21.5 - 5.77i)T + (5.40e3 - 3.12e3i)T^{2} \)
83 \( 1 + (33.9 + 58.8i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-82.3 - 22.0i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (40.6 + 40.6i)T + 9.40e3iT^{2} \)
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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

−12.02026646822183699716183636852, −11.14930294052149355044341594238, −9.950298738272519830002665761488, −8.642001675350450037333388317446, −7.71868017005765665900093716696, −6.38963391564451132212238842966, −6.00403624823033066091672258457, −4.35433823769966141688561854501, −3.89720511322426846831101763125, −2.50432102465396940148794130542, 0.60838978177694988239502307304, 3.05787485455263043272891799912, 3.85231885938074661295229486421, 4.53450320407055431421426184783, 5.97384421122294382803583147704, 6.81900008328750090252934624217, 8.186838328198351056281926132305, 8.972960711931182868028740915350, 10.68734637516475096808156849855, 11.26934001521315876134260832720

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