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Properties

Label 2-333-37.6-c2-0-11
Degree $2$
Conductor $333$
Sign $0.359 - 0.933i$
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  + (0.218 − 0.218i)2-s + 3.90i·4-s + (3.73 + 3.73i)5-s + 11.2·7-s + (1.72 + 1.72i)8-s + 1.63·10-s + 7.70i·11-s + (−9.81 − 9.81i)13-s + (2.45 − 2.45i)14-s − 14.8·16-s + (−14.2 − 14.2i)17-s + (10.4 + 10.4i)19-s + (−14.5 + 14.5i)20-s + (1.68 + 1.68i)22-s + (26.6 + 26.6i)23-s + ⋯
L(s)  = 1  + (0.109 − 0.109i)2-s + 0.976i·4-s + (0.747 + 0.747i)5-s + 1.60·7-s + (0.216 + 0.216i)8-s + 0.163·10-s + 0.700i·11-s + (−0.755 − 0.755i)13-s + (0.175 − 0.175i)14-s − 0.928·16-s + (−0.838 − 0.838i)17-s + (0.549 + 0.549i)19-s + (−0.729 + 0.729i)20-s + (0.0765 + 0.0765i)22-s + (1.15 + 1.15i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333\)    =    \(3^{2} \cdot 37\)
Sign: $0.359 - 0.933i$
Analytic conductor: \(9.07359\)
Root analytic conductor: \(3.01224\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{333} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 333,\ (\ :1),\ 0.359 - 0.933i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.83613 + 1.25988i\)
\(L(\frac12)\) \(\approx\) \(1.83613 + 1.25988i\)
\(L(2)\) 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 \( 1 + (32.4 - 17.7i)T \)
good2 \( 1 + (-0.218 + 0.218i)T - 4iT^{2} \)
5 \( 1 + (-3.73 - 3.73i)T + 25iT^{2} \)
7 \( 1 - 11.2T + 49T^{2} \)
11 \( 1 - 7.70iT - 121T^{2} \)
13 \( 1 + (9.81 + 9.81i)T + 169iT^{2} \)
17 \( 1 + (14.2 + 14.2i)T + 289iT^{2} \)
19 \( 1 + (-10.4 - 10.4i)T + 361iT^{2} \)
23 \( 1 + (-26.6 - 26.6i)T + 529iT^{2} \)
29 \( 1 + (0.147 - 0.147i)T - 841iT^{2} \)
31 \( 1 + (-13.0 + 13.0i)T - 961iT^{2} \)
41 \( 1 - 1.12iT - 1.68e3T^{2} \)
43 \( 1 + (26.2 + 26.2i)T + 1.84e3iT^{2} \)
47 \( 1 - 59.6T + 2.20e3T^{2} \)
53 \( 1 - 19.1T + 2.80e3T^{2} \)
59 \( 1 + (38.1 + 38.1i)T + 3.48e3iT^{2} \)
61 \( 1 + (2.53 - 2.53i)T - 3.72e3iT^{2} \)
67 \( 1 + 35.6iT - 4.48e3T^{2} \)
71 \( 1 + 133.T + 5.04e3T^{2} \)
73 \( 1 - 103. iT - 5.32e3T^{2} \)
79 \( 1 + (-3.44 - 3.44i)T + 6.24e3iT^{2} \)
83 \( 1 - 58.9T + 6.88e3T^{2} \)
89 \( 1 + (-7.78 + 7.78i)T - 7.92e3iT^{2} \)
97 \( 1 + (-48.8 - 48.8i)T + 9.40e3iT^{2} \)
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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

−11.56333969264584497460240204900, −10.75542203390532227385196771424, −9.736806974939848167479656861184, −8.618785516409298822446359287590, −7.56469780610610343097337637401, −7.06661867344994107643024294719, −5.38382700751692307310222915009, −4.54770484667010602488014783089, −2.97466020964597835409411615443, −1.94140659155795507912614870676, 1.12249657241072024381739579040, 2.14922853011817082263119156930, 4.60548591676106917539485594232, 5.02958414488032489905297415203, 6.08933987692327251795287797193, 7.24094468469190232273091218024, 8.717085938669091510538700856550, 9.060750512021526274247650224645, 10.40482607631076672687937143206, 11.03860701241691521428696177990

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