Properties

Label 2-3675-735.179-c0-0-1
Degree $2$
Conductor $3675$
Sign $0.626 + 0.779i$
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  + (0.294 − 0.955i)3-s + (0.988 − 0.149i)4-s + (0.563 + 0.826i)7-s + (−0.826 − 0.563i)9-s + (0.149 − 0.988i)12-s + (0.636 − 1.32i)13-s + (0.955 − 0.294i)16-s + (0.826 + 1.43i)19-s + (0.955 − 0.294i)21-s + (−0.781 + 0.623i)27-s + (0.680 + 0.733i)28-s + (−0.623 + 1.07i)31-s + (−0.900 − 0.433i)36-s + (0.108 − 0.722i)37-s + (−1.07 − 0.997i)39-s + ⋯
L(s)  = 1  + (0.294 − 0.955i)3-s + (0.988 − 0.149i)4-s + (0.563 + 0.826i)7-s + (−0.826 − 0.563i)9-s + (0.149 − 0.988i)12-s + (0.636 − 1.32i)13-s + (0.955 − 0.294i)16-s + (0.826 + 1.43i)19-s + (0.955 − 0.294i)21-s + (−0.781 + 0.623i)27-s + (0.680 + 0.733i)28-s + (−0.623 + 1.07i)31-s + (−0.900 − 0.433i)36-s + (0.108 − 0.722i)37-s + (−1.07 − 0.997i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.626 + 0.779i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.626 + 0.779i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.965576997\)
\(L(\frac12)\) \(\approx\) \(1.965576997\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.294 + 0.955i)T \)
5 \( 1 \)
7 \( 1 + (-0.563 - 0.826i)T \)
good2 \( 1 + (-0.988 + 0.149i)T^{2} \)
11 \( 1 + (-0.365 + 0.930i)T^{2} \)
13 \( 1 + (-0.636 + 1.32i)T + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (-0.733 + 0.680i)T^{2} \)
19 \( 1 + (-0.826 - 1.43i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.733 - 0.680i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.108 + 0.722i)T + (-0.955 - 0.294i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (1.75 + 0.400i)T + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.988 + 0.149i)T^{2} \)
53 \( 1 + (0.955 - 0.294i)T^{2} \)
59 \( 1 + (-0.0747 - 0.997i)T^{2} \)
61 \( 1 + (-1.95 - 0.294i)T + (0.955 + 0.294i)T^{2} \)
67 \( 1 + (1.65 + 0.955i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.728 + 0.0546i)T + (0.988 + 0.149i)T^{2} \)
79 \( 1 + (0.988 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.365 - 0.930i)T^{2} \)
97 \( 1 - 0.149iT - T^{2} \)
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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

−8.345032314914893306557444067793, −7.86395497887011076186724958817, −7.25178001945109497352819627976, −6.35141852375956223036730154975, −5.69175533314946324777841075505, −5.28457066110045806662681374446, −3.51923831889450914333401067450, −2.99114441166055704819776649375, −1.94868084683447675919625845531, −1.29159883436679084329789261202, 1.45751313645617828397654049947, 2.48942040418828431417264304109, 3.41451888279491963328730514603, 4.16147193690037436510421490913, 4.88284484108165708809668941561, 5.79519287739370142790497781696, 6.78550730553175025640354706409, 7.24648905633882935418608446707, 8.159177448601300809780023508958, 8.755985028603557278271076490103

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