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Properties

Label 2-675-9.7-c1-0-8
Degree $2$
Conductor $675$
Sign $0.468 + 0.883i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
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  + (0.816 − 1.41i)2-s + (−0.334 − 0.579i)4-s + (−0.252 + 0.437i)7-s + 2.17·8-s + (1.55 − 2.68i)11-s + (3.11 + 5.40i)13-s + (0.412 + 0.714i)14-s + (2.44 − 4.23i)16-s + 6.10·17-s − 5.57·19-s + (−2.53 − 4.38i)22-s + (−1.91 − 3.31i)23-s + 10.1·26-s + 0.338·28-s + (1.22 − 2.12i)29-s + ⋯
L(s)  = 1  + (0.577 − 1.00i)2-s + (−0.167 − 0.289i)4-s + (−0.0955 + 0.165i)7-s + 0.768·8-s + (0.467 − 0.809i)11-s + (0.865 + 1.49i)13-s + (0.110 + 0.191i)14-s + (0.611 − 1.05i)16-s + 1.47·17-s − 1.27·19-s + (−0.539 − 0.935i)22-s + (−0.398 − 0.690i)23-s + 1.99·26-s + 0.0638·28-s + (0.228 − 0.395i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.468 + 0.883i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.468 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99155 - 1.19879i\)
\(L(\frac12)\) \(\approx\) \(1.99155 - 1.19879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.816 + 1.41i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (0.252 - 0.437i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.55 + 2.68i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.11 - 5.40i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.10T + 17T^{2} \)
19 \( 1 + 5.57T + 19T^{2} \)
23 \( 1 + (1.91 + 3.31i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.22 + 2.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.11 + 3.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.72T + 37T^{2} \)
41 \( 1 + (2.72 + 4.71i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.663 - 1.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.85 - 3.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.54T + 53T^{2} \)
59 \( 1 + (-1.44 - 2.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.42 + 2.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.20 + 2.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.54T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 + (1.70 - 2.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.95 - 12.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.38T + 89T^{2} \)
97 \( 1 + (5.53 - 9.59i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.64959005855794997286613412941, −9.677923314062159304818355341391, −8.728016950908111940613932153422, −7.85252629486288838242300608392, −6.60145517430993275414912354410, −5.79616205720603944134846743175, −4.34797542282241678296371789210, −3.79120314296459164973731016058, −2.59822715710740763693091224356, −1.37835583599496737815986596781, 1.45397331447750814175227502396, 3.32415437551241882489602903441, 4.35700334228156191177975374346, 5.44607406712571464562750122742, 6.07180706138693132297772013820, 7.04550507345666796315112316615, 7.82325540759096991463163607376, 8.586669989319196956243284933142, 10.02012990002224084598284388083, 10.40770637264283679048926699902

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