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Properties

Label 2-3675-735.239-c0-0-0
Degree $2$
Conductor $3675$
Sign $-0.829 - 0.557i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.829 - 0.557i$
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: \((2,\ 3675,\ (\ :0),\ -0.829 - 0.557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.196019603\)
\(L(\frac12)\) \(\approx\) \(1.196019603\)
\(L(1)\) 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

−8.920970446023155260491016534516, −8.440065992584776729057242804090, −7.72597197220156270009658255075, −6.49917571088442790881811764890, −6.20916969187414949633967966112, −5.09829001590839374553122733177, −4.28023267703477699698761363843, −3.83189487559906568320559392267, −2.77260124546660505738832994933, −1.72519214609399642498350823683, 0.75754269722120365894077669200, 1.56876914023832586576509296601, 2.48217163497105725675259465371, 3.84118405222999124057758389626, 4.81210603622079533618583553160, 5.65678893570373154901467328475, 6.06980635830819015874178844036, 6.89917182919867120125055722415, 7.56575631904386443360493616340, 8.378878831324028263837812493460

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