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Properties

Label 2-165-165.98-c2-0-29
Degree $2$
Conductor $165$
Sign $0.646 + 0.763i$
Analytic cond. $4.49592$
Root an. cond. $2.12035$
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  + (−1.58 + 1.58i)2-s − 3i·3-s − 1.00i·4-s + (4 − 3i)5-s + (4.74 + 4.74i)6-s + (3.16 + 3.16i)7-s + (−4.74 − 4.74i)8-s − 9·9-s + (−1.58 + 11.0i)10-s + (−6.32 − 9i)11-s − 3.00·12-s + (3.16 − 3.16i)13-s − 10.0·14-s + (−9 − 12i)15-s + 19·16-s + (22.1 − 22.1i)17-s + ⋯
L(s)  = 1  + (−0.790 + 0.790i)2-s − i·3-s − 0.250i·4-s + (0.800 − 0.600i)5-s + (0.790 + 0.790i)6-s + (0.451 + 0.451i)7-s + (−0.592 − 0.592i)8-s − 9-s + (−0.158 + 1.10i)10-s + (−0.574 − 0.818i)11-s − 0.250·12-s + (0.243 − 0.243i)13-s − 0.714·14-s + (−0.599 − 0.800i)15-s + 1.18·16-s + (1.30 − 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.646 + 0.763i$
Analytic conductor: \(4.49592\)
Root analytic conductor: \(2.12035\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1),\ 0.646 + 0.763i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.900311 - 0.417440i\)
\(L(\frac12)\) \(\approx\) \(0.900311 - 0.417440i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 + (-4 + 3i)T \)
11 \( 1 + (6.32 + 9i)T \)
good2 \( 1 + (1.58 - 1.58i)T - 4iT^{2} \)
7 \( 1 + (-3.16 - 3.16i)T + 49iT^{2} \)
13 \( 1 + (-3.16 + 3.16i)T - 169iT^{2} \)
17 \( 1 + (-22.1 + 22.1i)T - 289iT^{2} \)
19 \( 1 + 12.6T + 361T^{2} \)
23 \( 1 + (-7 + 7i)T - 529iT^{2} \)
29 \( 1 + 18.9iT - 841T^{2} \)
31 \( 1 - 20T + 961T^{2} \)
37 \( 1 + (-7 + 7i)T - 1.36e3iT^{2} \)
41 \( 1 + 69.5T + 1.68e3T^{2} \)
43 \( 1 + (-22.1 + 22.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (-43 - 43i)T + 2.20e3iT^{2} \)
53 \( 1 + (17 - 17i)T - 2.80e3iT^{2} \)
59 \( 1 + 22T + 3.48e3T^{2} \)
61 \( 1 - 94.8iT - 3.72e3T^{2} \)
67 \( 1 + (47 - 47i)T - 4.48e3iT^{2} \)
71 \( 1 - 120iT - 5.04e3T^{2} \)
73 \( 1 + (-22.1 + 22.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 6.32T + 6.24e3T^{2} \)
83 \( 1 + (-60.0 - 60.0i)T + 6.88e3iT^{2} \)
89 \( 1 + 100T + 7.92e3T^{2} \)
97 \( 1 + (-43 + 43i)T - 9.40e3iT^{2} \)
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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

−12.52378820865159955030804084996, −11.67568950451279209473835234880, −10.15046318446987149350869889506, −8.905181008952890968472693064341, −8.337128410409393654057091160669, −7.40880470785479790227666813409, −6.14812999255925465432699556229, −5.38110680924042022607359655399, −2.75670455148768321508180349112, −0.827622736767975570358627154126, 1.81795124499749628997010588977, 3.31448897904670470354955182244, 5.00310121519156211031107316643, 6.16765574090880296724781382181, 7.934612484654441551376114892398, 9.056234145495604700355344475516, 10.11536891403089362849188380198, 10.39612213551611566092178622945, 11.20147307084552157999090217218, 12.43220254107025214568921032663

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