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Properties

Label 2-30e2-100.23-c1-0-21
Degree $2$
Conductor $900$
Sign $0.638 - 0.769i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  + (−1.41 − 0.0979i)2-s + (1.98 + 0.276i)4-s + (2.23 + 0.0570i)5-s + (1.87 + 1.87i)7-s + (−2.76 − 0.584i)8-s + (−3.14 − 0.299i)10-s + (−0.646 − 0.890i)11-s + (0.705 + 4.45i)13-s + (−2.45 − 2.82i)14-s + (3.84 + 1.09i)16-s + (2.44 + 1.24i)17-s + (0.659 + 2.03i)19-s + (4.41 + 0.731i)20-s + (0.825 + 1.31i)22-s + (−0.579 + 3.65i)23-s + ⋯
L(s)  = 1  + (−0.997 − 0.0692i)2-s + (0.990 + 0.138i)4-s + (0.999 + 0.0255i)5-s + (0.707 + 0.707i)7-s + (−0.978 − 0.206i)8-s + (−0.995 − 0.0947i)10-s + (−0.195 − 0.268i)11-s + (0.195 + 1.23i)13-s + (−0.656 − 0.754i)14-s + (0.961 + 0.273i)16-s + (0.593 + 0.302i)17-s + (0.151 + 0.465i)19-s + (0.986 + 0.163i)20-s + (0.175 + 0.281i)22-s + (−0.120 + 0.763i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.638 - 0.769i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.638 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16155 + 0.545724i\)
\(L(\frac12)\) \(\approx\) \(1.16155 + 0.545724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0979i)T \)
3 \( 1 \)
5 \( 1 + (-2.23 - 0.0570i)T \)
good7 \( 1 + (-1.87 - 1.87i)T + 7iT^{2} \)
11 \( 1 + (0.646 + 0.890i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.705 - 4.45i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-2.44 - 1.24i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.659 - 2.03i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.579 - 3.65i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (4.39 + 1.42i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.57 - 2.13i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.97 + 0.471i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (6.71 + 4.88i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-5.33 + 5.33i)T - 43iT^{2} \)
47 \( 1 + (1.62 - 0.829i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-0.482 + 0.245i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-5.72 - 4.16i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (8.21 - 5.96i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.62 - 5.14i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-13.1 - 4.27i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.81 - 0.921i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-4.83 + 14.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-7.15 - 3.64i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (1.03 + 1.42i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.93 + 3.80i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.11393728578283223922310364776, −9.235661196642742664903871869318, −8.828582435110636514775906394027, −7.83529777732483804548577854460, −6.94150337910679907135431051096, −5.90340011021020282896062483422, −5.34672831163964653551237805903, −3.64295178931372316839699264648, −2.22166932823535365616553132134, −1.55327732615884197136792993710, 0.895844430448754165995086798003, 2.07969762455334968969965567202, 3.25452011354427873624844015052, 4.94120374367672140633302956195, 5.74656138810128350167252321244, 6.73251594746445600033028273001, 7.66174072679969986126689376903, 8.218773747372919388321735833633, 9.350165078629660615800504613790, 9.872113836789207405630014054349

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