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Properties

Label 2-18e2-324.275-c1-0-25
Degree $2$
Conductor $324$
Sign $0.490 - 0.871i$
Analytic cond. $2.58715$
Root an. cond. $1.60846$
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  + (1.40 + 0.114i)2-s + (−0.807 + 1.53i)3-s + (1.97 + 0.323i)4-s + (1.37 + 0.593i)5-s + (−1.31 + 2.06i)6-s + (0.328 − 0.0191i)7-s + (2.74 + 0.682i)8-s + (−1.69 − 2.47i)9-s + (1.87 + 0.995i)10-s + (−0.378 + 0.509i)11-s + (−2.08 + 2.76i)12-s + (−0.663 + 0.703i)13-s + (0.464 + 0.0107i)14-s + (−2.02 + 1.63i)15-s + (3.79 + 1.27i)16-s + (−2.30 − 0.406i)17-s + ⋯
L(s)  = 1  + (0.996 + 0.0811i)2-s + (−0.466 + 0.884i)3-s + (0.986 + 0.161i)4-s + (0.615 + 0.265i)5-s + (−0.536 + 0.843i)6-s + (0.124 − 0.00722i)7-s + (0.970 + 0.241i)8-s + (−0.565 − 0.824i)9-s + (0.592 + 0.314i)10-s + (−0.114 + 0.153i)11-s + (−0.603 + 0.797i)12-s + (−0.184 + 0.195i)13-s + (0.124 + 0.00286i)14-s + (−0.521 + 0.420i)15-s + (0.947 + 0.319i)16-s + (−0.559 − 0.0985i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.490 - 0.871i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.490 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92915 + 1.12864i\)
\(L(\frac12)\) \(\approx\) \(1.92915 + 1.12864i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.114i)T \)
3 \( 1 + (0.807 - 1.53i)T \)
good5 \( 1 + (-1.37 - 0.593i)T + (3.43 + 3.63i)T^{2} \)
7 \( 1 + (-0.328 + 0.0191i)T + (6.95 - 0.812i)T^{2} \)
11 \( 1 + (0.378 - 0.509i)T + (-3.15 - 10.5i)T^{2} \)
13 \( 1 + (0.663 - 0.703i)T + (-0.755 - 12.9i)T^{2} \)
17 \( 1 + (2.30 + 0.406i)T + (15.9 + 5.81i)T^{2} \)
19 \( 1 + (-0.442 + 0.0779i)T + (17.8 - 6.49i)T^{2} \)
23 \( 1 + (0.169 - 2.91i)T + (-22.8 - 2.67i)T^{2} \)
29 \( 1 + (0.405 + 1.71i)T + (-25.9 + 13.0i)T^{2} \)
31 \( 1 + (2.24 + 4.47i)T + (-18.5 + 24.8i)T^{2} \)
37 \( 1 + (-6.68 + 2.43i)T + (28.3 - 23.7i)T^{2} \)
41 \( 1 + (-0.549 + 0.164i)T + (34.2 - 22.5i)T^{2} \)
43 \( 1 + (-1.06 + 9.07i)T + (-41.8 - 9.91i)T^{2} \)
47 \( 1 + (-2.99 - 1.50i)T + (28.0 + 37.6i)T^{2} \)
53 \( 1 + (7.54 + 4.35i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.60 - 4.83i)T + (-16.9 + 56.5i)T^{2} \)
61 \( 1 + (4.02 - 2.64i)T + (24.1 - 56.0i)T^{2} \)
67 \( 1 + (-3.29 + 13.8i)T + (-59.8 - 30.0i)T^{2} \)
71 \( 1 + (0.732 + 0.614i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-8.47 + 7.11i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (11.2 + 3.38i)T + (66.0 + 43.4i)T^{2} \)
83 \( 1 + (4.80 - 16.0i)T + (-69.3 - 45.6i)T^{2} \)
89 \( 1 + (4.53 + 5.40i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (-4.10 - 9.52i)T + (-66.5 + 70.5i)T^{2} \)
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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

−11.66929467378487314921973322073, −11.01939215594438567288425109464, −10.11445203469929168094621704462, −9.255076730900866239369873266405, −7.74182876480854088656767217511, −6.50001290243726210896905493251, −5.73477434261757360923009560513, −4.75746039975982661017382391296, −3.74714722575311324489673342195, −2.32715515627298679236297228323, 1.55994678298483920804154150476, 2.82769583273312262018633459446, 4.60205483620450938122899751644, 5.57102032108428017839616412433, 6.35655275148101727398582544690, 7.31407926555172041791400468394, 8.371074490035275697481102599342, 9.837725944768587942344056898227, 10.96022806265510264105794868943, 11.54971693010077082424231368548

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