Properties

Label 2-3450-1.1-c1-0-12
Degree $2$
Conductor $3450$
Sign $1$
Analytic cond. $27.5483$
Root an. cond. $5.24865$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 3·7-s − 8-s + 9-s + 5.77·11-s + 12-s − 3.77·13-s + 3·14-s + 16-s + 0.772·17-s − 18-s + 7.77·19-s − 3·21-s − 5.77·22-s + 23-s − 24-s + 3.77·26-s + 27-s − 3·28-s + 3·29-s − 9.54·31-s − 32-s + 5.77·33-s − 0.772·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 0.333·9-s + 1.74·11-s + 0.288·12-s − 1.04·13-s + 0.801·14-s + 0.250·16-s + 0.187·17-s − 0.235·18-s + 1.78·19-s − 0.654·21-s − 1.23·22-s + 0.208·23-s − 0.204·24-s + 0.739·26-s + 0.192·27-s − 0.566·28-s + 0.557·29-s − 1.71·31-s − 0.176·32-s + 1.00·33-s − 0.132·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(27.5483\)
Root analytic conductor: \(5.24865\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.573499019\)
\(L(\frac12)\) \(\approx\) \(1.573499019\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 \)
23 \( 1 - T \)
good7 \( 1 + 3T + 7T^{2} \)
11 \( 1 - 5.77T + 11T^{2} \)
13 \( 1 + 3.77T + 13T^{2} \)
17 \( 1 - 0.772T + 17T^{2} \)
19 \( 1 - 7.77T + 19T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 9.54T + 31T^{2} \)
37 \( 1 - 6.77T + 37T^{2} \)
41 \( 1 + 5.77T + 41T^{2} \)
43 \( 1 - 7.77T + 43T^{2} \)
47 \( 1 + 8.77T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 9.54T + 61T^{2} \)
67 \( 1 + 9.54T + 67T^{2} \)
71 \( 1 + 4.77T + 71T^{2} \)
73 \( 1 - 6.54T + 73T^{2} \)
79 \( 1 - 2.22T + 79T^{2} \)
83 \( 1 + T + 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 - 17.5T + 97T^{2} \)
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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

−8.952548758755425993860558627203, −7.72532875423666034563636185527, −7.25898045343318467564135063167, −6.58627516889358795843683636662, −5.82512385636807742735454555055, −4.69059877759759686647558142392, −3.53930909040159474666659532958, −3.11572604966901884782002356794, −1.92510402792346662634850703788, −0.815943988702373896305767237976, 0.815943988702373896305767237976, 1.92510402792346662634850703788, 3.11572604966901884782002356794, 3.53930909040159474666659532958, 4.69059877759759686647558142392, 5.82512385636807742735454555055, 6.58627516889358795843683636662, 7.25898045343318467564135063167, 7.72532875423666034563636185527, 8.952548758755425993860558627203

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