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Properties

Label 2-30e2-100.23-c1-0-36
Degree $2$
Conductor $900$
Sign $0.817 - 0.576i$
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  + (−0.618 + 1.27i)2-s + (−1.23 − 1.57i)4-s + (2.17 − 0.500i)5-s + (−0.974 − 0.974i)7-s + (2.76 − 0.595i)8-s + (−0.711 + 3.08i)10-s + (1.28 + 1.77i)11-s + (0.471 + 2.97i)13-s + (1.84 − 0.636i)14-s + (−0.953 + 3.88i)16-s + (0.167 + 0.0855i)17-s + (−2.61 − 8.06i)19-s + (−3.47 − 2.81i)20-s + (−3.04 + 0.540i)22-s + (−0.127 + 0.802i)23-s + ⋯
L(s)  = 1  + (−0.437 + 0.899i)2-s + (−0.617 − 0.786i)4-s + (0.974 − 0.224i)5-s + (−0.368 − 0.368i)7-s + (0.977 − 0.210i)8-s + (−0.224 + 0.974i)10-s + (0.387 + 0.533i)11-s + (0.130 + 0.826i)13-s + (0.492 − 0.170i)14-s + (−0.238 + 0.971i)16-s + (0.0407 + 0.0207i)17-s + (−0.600 − 1.84i)19-s + (−0.777 − 0.628i)20-s + (−0.649 + 0.115i)22-s + (−0.0264 + 0.167i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.817 - 0.576i$
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: \((2,\ 900,\ (\ :1/2),\ 0.817 - 0.576i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34155 + 0.425567i\)
\(L(\frac12)\) \(\approx\) \(1.34155 + 0.425567i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.618 - 1.27i)T \)
3 \( 1 \)
5 \( 1 + (-2.17 + 0.500i)T \)
good7 \( 1 + (0.974 + 0.974i)T + 7iT^{2} \)
11 \( 1 + (-1.28 - 1.77i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.471 - 2.97i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-0.167 - 0.0855i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (2.61 + 8.06i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.127 - 0.802i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-9.37 - 3.04i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.18 + 1.35i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.96 + 1.10i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-2.94 - 2.13i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-1.86 + 1.86i)T - 43iT^{2} \)
47 \( 1 + (-5.26 + 2.68i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (6.90 - 3.51i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-4.74 - 3.44i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.78 - 2.02i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.24 - 6.37i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (9.69 + 3.15i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.675 - 0.106i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-3.74 + 11.5i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-6.86 - 3.49i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (4.99 + 6.87i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.10 - 10.0i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.01320456449300337595669146425, −9.113419127299011426350001881606, −8.819168191176694236801380270392, −7.49217050708243679998259413075, −6.62170077008818817032201345384, −6.24657707546452886033537831536, −4.93200963397160234783313608672, −4.33132277298159845095498766074, −2.45714919107679564474897396889, −1.02390938049596579422891136698, 1.14076735312036848559820095663, 2.45759036372337649601556954618, 3.27913860148971590669263668128, 4.49020916029612541823527517238, 5.80520819154123507992458665980, 6.39353468927249513802973373755, 7.85817919070051953085948654494, 8.500630821460245282215968381391, 9.407345980198549673744160232390, 10.12979928297385538365501922851

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