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Properties

Label 2-675-9.5-c2-0-26
Degree $2$
Conductor $675$
Sign $0.277 + 0.960i$
Analytic cond. $18.3924$
Root an. cond. $4.28863$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.15 − 0.668i)2-s + (−1.10 + 1.91i)4-s + (−4.10 − 7.10i)7-s + 8.30i·8-s + (5.67 − 3.27i)11-s + (−0.749 + 1.29i)13-s + (−9.50 − 5.49i)14-s + (1.13 + 1.97i)16-s − 15.1i·17-s + 25.9·19-s + (4.38 − 7.58i)22-s + (−20.1 − 11.6i)23-s + 2.00i·26-s + 18.1·28-s + (6.96 − 4.02i)29-s + ⋯
L(s)  = 1  + (0.579 − 0.334i)2-s + (−0.276 + 0.478i)4-s + (−0.586 − 1.01i)7-s + 1.03i·8-s + (0.515 − 0.297i)11-s + (−0.0576 + 0.0998i)13-s + (−0.679 − 0.392i)14-s + (0.0710 + 0.123i)16-s − 0.889i·17-s + 1.36·19-s + (0.199 − 0.344i)22-s + (−0.873 − 0.504i)23-s + 0.0771i·26-s + 0.647·28-s + (0.240 − 0.138i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(18.3924\)
Root analytic conductor: \(4.28863\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (476, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1),\ 0.277 + 0.960i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.911513575\)
\(L(\frac12)\) \(\approx\) \(1.911513575\)
\(L(2)\) 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 \( 1 + (-1.15 + 0.668i)T + (2 - 3.46i)T^{2} \)
7 \( 1 + (4.10 + 7.10i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-5.67 + 3.27i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (0.749 - 1.29i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 15.1iT - 289T^{2} \)
19 \( 1 - 25.9T + 361T^{2} \)
23 \( 1 + (20.1 + 11.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-6.96 + 4.02i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-22.5 + 38.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 62.8T + 1.36e3T^{2} \)
41 \( 1 + (9.97 + 5.75i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (21.3 + 36.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-14.2 + 8.25i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 66.0iT - 2.80e3T^{2} \)
59 \( 1 + (-0.373 - 0.215i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-15.7 - 27.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-47.9 + 83.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 84.2iT - 5.04e3T^{2} \)
73 \( 1 - 63.5T + 5.32e3T^{2} \)
79 \( 1 + (9.06 + 15.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (87.4 - 50.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 86.3iT - 7.92e3T^{2} \)
97 \( 1 + (34.5 + 59.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.02237790000533282732533723106, −9.459755757960132110556433718961, −8.288159365356896722150846262775, −7.50056309926879618795354670363, −6.57184919219075176555493084538, −5.39158086097710134607988385679, −4.29258308266352057554004458891, −3.61585075836221266151573590522, −2.55788225252576684173263425729, −0.63567369491801050999517334574, 1.31828515013168977544875717096, 2.95426268314180189361457728693, 4.07543229799293666005546479386, 5.14796761249529624917463669947, 5.97340786200493665483998934271, 6.56967815436670962019758494833, 7.77648642406345426546666234797, 8.914853291064190090899898371583, 9.626176890147571438541468939845, 10.22388850498007409019948867626

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