Properties

Label 2-675-135.83-c1-0-19
Degree $2$
Conductor $675$
Sign $-0.230 - 0.973i$
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  + (−0.490 + 0.343i)2-s + (1.72 − 0.150i)3-s + (−0.561 + 1.54i)4-s + (−0.794 + 0.665i)6-s + (0.0636 − 0.136i)7-s + (−0.563 − 2.10i)8-s + (2.95 − 0.518i)9-s + (−3.54 + 4.23i)11-s + (−0.737 + 2.74i)12-s + (−1.08 + 1.54i)13-s + (0.0156 + 0.0887i)14-s + (−1.51 − 1.27i)16-s + (−1.18 + 4.40i)17-s + (−1.27 + 1.26i)18-s + (6.55 + 3.78i)19-s + ⋯
L(s)  = 1  + (−0.346 + 0.242i)2-s + (0.996 − 0.0866i)3-s + (−0.280 + 0.771i)4-s + (−0.324 + 0.271i)6-s + (0.0240 − 0.0515i)7-s + (−0.199 − 0.744i)8-s + (0.984 − 0.172i)9-s + (−1.07 + 1.27i)11-s + (−0.212 + 0.792i)12-s + (−0.300 + 0.429i)13-s + (0.00418 + 0.0237i)14-s + (−0.379 − 0.318i)16-s + (−0.286 + 1.06i)17-s + (−0.299 + 0.298i)18-s + (1.50 + 0.867i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.230 - 0.973i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.230 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.885161 + 1.11878i\)
\(L(\frac12)\) \(\approx\) \(0.885161 + 1.11878i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 + 0.150i)T \)
5 \( 1 \)
good2 \( 1 + (0.490 - 0.343i)T + (0.684 - 1.87i)T^{2} \)
7 \( 1 + (-0.0636 + 0.136i)T + (-4.49 - 5.36i)T^{2} \)
11 \( 1 + (3.54 - 4.23i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.08 - 1.54i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (1.18 - 4.40i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.55 - 3.78i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.883 + 0.412i)T + (14.7 - 17.6i)T^{2} \)
29 \( 1 + (0.486 - 2.75i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.70 - 0.622i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (5.71 + 1.53i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.682 - 0.120i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (0.151 + 1.73i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (10.0 + 4.70i)T + (30.2 + 36.0i)T^{2} \)
53 \( 1 + (0.204 - 0.204i)T - 53iT^{2} \)
59 \( 1 + (-0.796 + 0.668i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-9.12 + 3.32i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-8.96 - 6.27i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (2.53 - 1.46i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.95 + 2.66i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (11.9 + 2.10i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (4.10 + 5.85i)T + (-28.3 + 77.9i)T^{2} \)
89 \( 1 + (-6.74 + 11.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.56 - 0.311i)T + (95.5 - 16.8i)T^{2} \)
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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

−10.26505142487407071674985761022, −9.811485794953446422675969335170, −8.879368623097581787269249056752, −8.099485900534193749656025053015, −7.45782928698606171331310720636, −6.81197739515112441712874806882, −5.11463208808947909804048253792, −4.07673043099270537126935886435, −3.13235611301918025425986285773, −1.88740920747129897678880480409, 0.77267897371901376553990456622, 2.45548450977711912822693873613, 3.24769196843846900081626173246, 4.88203643442244975708062381079, 5.47023320054553718937136330435, 6.89434282993420651303663734341, 7.931410366484697697475694404022, 8.605300042296866862324606887509, 9.485789967916657118174454650688, 10.00304034977832812841456053717

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