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Properties

Label 2-3675-735.239-c0-0-1
Degree $2$
Conductor $3675$
Sign $-0.944 + 0.329i$
Analytic cond. $1.83406$
Root an. cond. $1.35427$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.433 − 0.900i)3-s + (0.222 + 0.974i)4-s + (−0.781 − 0.623i)7-s + (−0.623 − 0.781i)9-s + (0.974 + 0.222i)12-s + (−1.40 − 1.12i)13-s + (−0.900 + 0.433i)16-s − 1.24·19-s + (−0.900 + 0.433i)21-s + (−0.974 + 0.222i)27-s + (0.433 − 0.900i)28-s − 0.445·31-s + (0.623 − 0.781i)36-s + (−0.433 − 0.0990i)37-s + (−1.62 + 0.781i)39-s + ⋯
L(s)  = 1  + (0.433 − 0.900i)3-s + (0.222 + 0.974i)4-s + (−0.781 − 0.623i)7-s + (−0.623 − 0.781i)9-s + (0.974 + 0.222i)12-s + (−1.40 − 1.12i)13-s + (−0.900 + 0.433i)16-s − 1.24·19-s + (−0.900 + 0.433i)21-s + (−0.974 + 0.222i)27-s + (0.433 − 0.900i)28-s − 0.445·31-s + (0.623 − 0.781i)36-s + (−0.433 − 0.0990i)37-s + (−1.62 + 0.781i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.944 + 0.329i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (974, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.944 + 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5339665837\)
\(L(\frac12)\) \(\approx\) \(0.5339665837\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.433 + 0.900i)T \)
5 \( 1 \)
7 \( 1 + (0.781 + 0.623i)T \)
good2 \( 1 + (-0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.222 + 0.974i)T^{2} \)
13 \( 1 + (1.40 + 1.12i)T + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (-0.900 - 0.433i)T^{2} \)
19 \( 1 + 1.24T + T^{2} \)
23 \( 1 + (-0.900 + 0.433i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + 0.445T + T^{2} \)
37 \( 1 + (0.433 + 0.0990i)T + (0.900 + 0.433i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (0.541 + 1.12i)T + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.222 - 0.974i)T^{2} \)
53 \( 1 + (-0.900 + 0.433i)T^{2} \)
59 \( 1 + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
67 \( 1 - 1.80iT - T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.347 + 0.277i)T + (0.222 - 0.974i)T^{2} \)
79 \( 1 - 0.445T + T^{2} \)
83 \( 1 + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.222 - 0.974i)T^{2} \)
97 \( 1 + 1.24iT - T^{2} \)
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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

−8.278305108710086821979838351181, −7.50483146267765188031201730384, −7.11114952289981818551825967622, −6.47606025150969044650464974587, −5.51459822127406712389316182707, −4.32809827453038452931133496782, −3.46420195374221534441148103049, −2.80661934363568333283845742314, −2.01708771530275695965787816868, −0.25166849713134337094959157966, 2.00916646777056426328183589611, 2.52359508324462114874456403445, 3.64276616357900361865269034685, 4.66113733127084381286865521434, 5.08936620721014171071715365239, 6.10360336062646074834830805994, 6.62917632758595162701945708053, 7.54328277203507987370014837003, 8.626735050165523204415649707793, 9.219307409046803569940206265967

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