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Properties

Label 2-165-165.29-c1-0-19
Degree $2$
Conductor $165$
Sign $-0.898 + 0.438i$
Analytic cond. $1.31753$
Root an. cond. $1.14783$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.55 − 2.14i)2-s + (−1.61 + 0.637i)3-s + (−1.54 − 4.76i)4-s + (−1.59 − 1.56i)5-s + (−1.14 + 4.44i)6-s + (0.391 + 1.20i)7-s + (−7.57 − 2.46i)8-s + (2.18 − 2.05i)9-s + (−5.83 + 0.992i)10-s + (2.72 − 1.89i)11-s + (5.52 + 6.68i)12-s + (2.33 + 1.69i)13-s + (3.19 + 1.03i)14-s + (3.57 + 1.49i)15-s + (−8.94 + 6.50i)16-s + (−2.46 − 3.39i)17-s + ⋯
L(s)  = 1  + (1.10 − 1.51i)2-s + (−0.929 + 0.367i)3-s + (−0.773 − 2.38i)4-s + (−0.715 − 0.698i)5-s + (−0.466 + 1.81i)6-s + (0.148 + 0.455i)7-s + (−2.67 − 0.869i)8-s + (0.729 − 0.684i)9-s + (−1.84 + 0.313i)10-s + (0.821 − 0.570i)11-s + (1.59 + 1.92i)12-s + (0.648 + 0.471i)13-s + (0.853 + 0.277i)14-s + (0.922 + 0.386i)15-s + (−2.23 + 1.62i)16-s + (−0.598 − 0.824i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.898 + 0.438i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -0.898 + 0.438i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.292510 - 1.26582i\)
\(L(\frac12)\) \(\approx\) \(0.292510 - 1.26582i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.61 - 0.637i)T \)
5 \( 1 + (1.59 + 1.56i)T \)
11 \( 1 + (-2.72 + 1.89i)T \)
good2 \( 1 + (-1.55 + 2.14i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + (-0.391 - 1.20i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.33 - 1.69i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.46 + 3.39i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.26 - 0.735i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.33T + 23T^{2} \)
29 \( 1 + (0.920 + 2.83i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.30 - 2.40i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (4.41 - 1.43i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.88 - 8.87i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.388T + 43T^{2} \)
47 \( 1 + (1.53 - 4.73i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.20 + 2.32i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.853 + 0.277i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.883 + 1.21i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 11.9iT - 67T^{2} \)
71 \( 1 + (-6.06 - 8.35i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.481 - 1.48i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.69 - 9.21i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.69 + 2.32i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.36iT - 89T^{2} \)
97 \( 1 + (5.61 - 7.73i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.11568440426842321741296103658, −11.48219651904633309431181441490, −11.10782417537461651823309013288, −9.678954247915824214242237401148, −8.849118293281185452914754489105, −6.52043835495876492955876003562, −5.28545388308942957806227847361, −4.49086269108010948682462091179, −3.41553239754769167596354336919, −1.13959044655828765631750794060, 3.67253077083957222606941818797, 4.66414568160121586055351733097, 5.94019466420233021916170957892, 6.90776153091540420949847873982, 7.37659750211068643545525153087, 8.601495523765967790219421351256, 10.60600863174133289710146971175, 11.62664155711630133543759419644, 12.50260536466971901649764266220, 13.37170255203274712058346260756

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