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Properties

Label 2-675-15.2-c1-0-6
Degree $2$
Conductor $675$
Sign $-0.229 - 0.973i$
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  + 2i·4-s + (3.67 + 3.67i)7-s + (−3.67 + 3.67i)13-s − 4·16-s − 7i·19-s + (−7.34 + 7.34i)28-s + 4·31-s + (3.67 + 3.67i)37-s + (−7.34 + 7.34i)43-s + 20i·49-s + (−7.34 − 7.34i)52-s − 61-s − 8i·64-s + (11.0 + 11.0i)67-s + (11.0 − 11.0i)73-s + ⋯
L(s)  = 1  + i·4-s + (1.38 + 1.38i)7-s + (−1.01 + 1.01i)13-s − 16-s − 1.60i·19-s + (−1.38 + 1.38i)28-s + 0.718·31-s + (0.604 + 0.604i)37-s + (−1.12 + 1.12i)43-s + 2.85i·49-s + (−1.01 − 1.01i)52-s − 0.128·61-s − i·64-s + (1.34 + 1.34i)67-s + (1.29 − 1.29i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.953093 + 1.20428i\)
\(L(\frac12)\) \(\approx\) \(0.953093 + 1.20428i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT^{2} \)
7 \( 1 + (-3.67 - 3.67i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (3.67 - 3.67i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 7iT - 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3.67 - 3.67i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (7.34 - 7.34i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + (-11.0 - 11.0i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-11.0 + 11.0i)T - 73iT^{2} \)
79 \( 1 + 17iT - 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-3.67 - 3.67i)T + 97iT^{2} \)
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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

−11.16683856603533676347472487587, −9.604016818469795162323529829922, −8.880651150672012664758157605881, −8.234078066208530480228806102945, −7.39017687193380375808236301925, −6.41299073224123257606049838817, −4.95045381326338290532803638602, −4.56720212325854621357988456073, −2.85955783127722911518518210095, −2.06119691997306733241214824942, 0.843625148022458331575679911331, 2.02970496851589412391352550230, 3.84393010601952794038943619774, 4.86405019531050475328461314647, 5.51666933967650530245093631763, 6.77094855373887078832192123822, 7.70585472803206203667370203725, 8.291323182298894291246853307458, 9.770774223069268770939321785292, 10.27306934209872868822021492634

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