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Properties

Label 2-675-15.14-c2-0-18
Degree $2$
Conductor $675$
Sign $0.894 - 0.447i$
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  + 1.25·2-s − 2.43·4-s − 7.62i·7-s − 8.05·8-s + 15.8i·11-s + 10.7i·13-s − 9.54i·14-s − 0.352·16-s − 1.55·17-s + 23.9·19-s + 19.7i·22-s + 33.5·23-s + 13.3i·26-s + 18.5i·28-s − 22.0i·29-s + ⋯
L(s)  = 1  + 0.626·2-s − 0.608·4-s − 1.08i·7-s − 1.00·8-s + 1.43i·11-s + 0.823i·13-s − 0.681i·14-s − 0.0220·16-s − 0.0916·17-s + 1.26·19-s + 0.899i·22-s + 1.45·23-s + 0.515i·26-s + 0.662i·28-s − 0.760i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(18.3924\)
Root analytic conductor: \(4.28863\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1),\ 0.894 - 0.447i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.946693312\)
\(L(\frac12)\) \(\approx\) \(1.946693312\)
\(L(2)\) 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 \( 1 - 1.25T + 4T^{2} \)
7 \( 1 + 7.62iT - 49T^{2} \)
11 \( 1 - 15.8iT - 121T^{2} \)
13 \( 1 - 10.7iT - 169T^{2} \)
17 \( 1 + 1.55T + 289T^{2} \)
19 \( 1 - 23.9T + 361T^{2} \)
23 \( 1 - 33.5T + 529T^{2} \)
29 \( 1 + 22.0iT - 841T^{2} \)
31 \( 1 - 27.0T + 961T^{2} \)
37 \( 1 + 28.7iT - 1.36e3T^{2} \)
41 \( 1 - 48.2iT - 1.68e3T^{2} \)
43 \( 1 - 9.02iT - 1.84e3T^{2} \)
47 \( 1 - 48.8T + 2.20e3T^{2} \)
53 \( 1 - 59.6T + 2.80e3T^{2} \)
59 \( 1 - 99.8iT - 3.48e3T^{2} \)
61 \( 1 + 89.2T + 3.72e3T^{2} \)
67 \( 1 + 1.85iT - 4.48e3T^{2} \)
71 \( 1 - 126. iT - 5.04e3T^{2} \)
73 \( 1 - 101. iT - 5.32e3T^{2} \)
79 \( 1 + 108.T + 6.24e3T^{2} \)
83 \( 1 - 3.54T + 6.88e3T^{2} \)
89 \( 1 + 108. iT - 7.92e3T^{2} \)
97 \( 1 - 88.8iT - 9.40e3T^{2} \)
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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

−10.17352321004004642487944570905, −9.612400504373854449202878441203, −8.768639853996972927873065793594, −7.45826607204768169904584131474, −6.94903846726114205702239810075, −5.65801118533666241537744921202, −4.54447362603035059671436112154, −4.17586407491778984520416958388, −2.83148223521364974981028504100, −1.07037872506906410493049207152, 0.75403634860828787160398151040, 2.87866505173137486269135422764, 3.43534324590974452818725442353, 4.99145152737033553332871741962, 5.49926953568599829198767025438, 6.30873617152289144606193652424, 7.71957984185506437030975896083, 8.823706178517922187763244190876, 9.001905549552400984190342828836, 10.25632320826823512215860431440

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