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Properties

Label 2-330-55.13-c1-0-7
Degree $2$
Conductor $330$
Sign $0.722 + 0.691i$
Analytic cond. $2.63506$
Root an. cond. $1.62328$
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  + (0.891 − 0.453i)2-s + (−0.987 − 0.156i)3-s + (0.587 − 0.809i)4-s + (2.07 − 0.830i)5-s + (−0.951 + 0.309i)6-s + (0.320 + 2.02i)7-s + (0.156 − 0.987i)8-s + (0.951 + 0.309i)9-s + (1.47 − 1.68i)10-s + (3.31 − 0.199i)11-s + (−0.707 + 0.707i)12-s + (−2.30 − 4.52i)13-s + (1.20 + 1.65i)14-s + (−2.18 + 0.495i)15-s + (−0.309 − 0.951i)16-s + (−1.46 + 2.86i)17-s + ⋯
L(s)  = 1  + (0.630 − 0.321i)2-s + (−0.570 − 0.0903i)3-s + (0.293 − 0.404i)4-s + (0.928 − 0.371i)5-s + (−0.388 + 0.126i)6-s + (0.121 + 0.764i)7-s + (0.0553 − 0.349i)8-s + (0.317 + 0.103i)9-s + (0.465 − 0.531i)10-s + (0.998 − 0.0601i)11-s + (−0.204 + 0.204i)12-s + (−0.639 − 1.25i)13-s + (0.321 + 0.443i)14-s + (−0.563 + 0.127i)15-s + (−0.0772 − 0.237i)16-s + (−0.354 + 0.696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ 0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72423 - 0.692055i\)
\(L(\frac12)\) \(\approx\) \(1.72423 - 0.692055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.891 + 0.453i)T \)
3 \( 1 + (0.987 + 0.156i)T \)
5 \( 1 + (-2.07 + 0.830i)T \)
11 \( 1 + (-3.31 + 0.199i)T \)
good7 \( 1 + (-0.320 - 2.02i)T + (-6.65 + 2.16i)T^{2} \)
13 \( 1 + (2.30 + 4.52i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (1.46 - 2.86i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (-3.80 + 2.76i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.56 - 1.56i)T + 23iT^{2} \)
29 \( 1 + (7.79 + 5.66i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.67 - 8.23i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.102 - 0.0163i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-3.05 - 4.20i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-0.380 + 0.380i)T - 43iT^{2} \)
47 \( 1 + (1.57 - 9.93i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (11.9 - 6.10i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (7.56 - 10.4i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.38 + 1.42i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (0.814 - 0.814i)T - 67iT^{2} \)
71 \( 1 + (1.18 + 3.65i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (14.0 - 2.22i)T + (69.4 - 22.5i)T^{2} \)
79 \( 1 + (-2.32 + 7.15i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.27 + 1.66i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 - 7.49iT - 89T^{2} \)
97 \( 1 + (6.40 + 12.5i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.59229884771025996837170659281, −10.72554726202609716505916853551, −9.681390132958509806598177477954, −8.957568614557132169798825954903, −7.45216503143463861187242028193, −6.14567800431482272936504338083, −5.57873857132394831583615132088, −4.62161350786647279296806271213, −2.94886502132055641641883816251, −1.49128101231452200004070005681, 1.85440108770388491587331485225, 3.67238272892223549110323910565, 4.76681757364988169846500087642, 5.81134480332844770181938434743, 6.84828661743328086229595431061, 7.32163705895766704216310729168, 9.173010682327557146877939267461, 9.792193533238539206663710534101, 11.03773711072982616777054474122, 11.60753558304791597437753119863

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