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Properties

Label 2-2475-2475.1429-c0-0-0
Degree $2$
Conductor $2475$
Sign $-0.335 + 0.942i$
Analytic cond. $1.23518$
Root an. cond. $1.11138$
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  + (0.978 − 0.207i)3-s + (0.104 − 0.994i)4-s − 5-s + (0.913 − 0.406i)9-s + (−0.669 − 0.743i)11-s + (−0.104 − 0.994i)12-s + (−0.978 + 0.207i)15-s + (−0.978 − 0.207i)16-s + (−0.104 + 0.994i)20-s + (−0.244 − 1.14i)23-s + 25-s + (0.809 − 0.587i)27-s + (−1.22 − 0.544i)31-s + (−0.809 − 0.587i)33-s + (−0.309 − 0.951i)36-s + (1.64 − 0.535i)37-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)3-s + (0.104 − 0.994i)4-s − 5-s + (0.913 − 0.406i)9-s + (−0.669 − 0.743i)11-s + (−0.104 − 0.994i)12-s + (−0.978 + 0.207i)15-s + (−0.978 − 0.207i)16-s + (−0.104 + 0.994i)20-s + (−0.244 − 1.14i)23-s + 25-s + (0.809 − 0.587i)27-s + (−1.22 − 0.544i)31-s + (−0.809 − 0.587i)33-s + (−0.309 − 0.951i)36-s + (1.64 − 0.535i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.335 + 0.942i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ -0.335 + 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.272402568\)
\(L(\frac12)\) \(\approx\) \(1.272402568\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + T \)
11 \( 1 + (0.669 + 0.743i)T \)
good2 \( 1 + (-0.104 + 0.994i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.104 - 0.994i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.244 + 1.14i)T + (-0.913 + 0.406i)T^{2} \)
29 \( 1 + (-0.669 + 0.743i)T^{2} \)
31 \( 1 + (1.22 + 0.544i)T + (0.669 + 0.743i)T^{2} \)
37 \( 1 + (-1.64 + 0.535i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.104 + 0.994i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.704 + 1.58i)T + (-0.669 + 0.743i)T^{2} \)
53 \( 1 + (0.873 - 1.20i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (1.30 - 1.45i)T + (-0.104 - 0.994i)T^{2} \)
61 \( 1 + (0.104 - 0.994i)T^{2} \)
67 \( 1 + (-0.809 + 1.81i)T + (-0.669 - 0.743i)T^{2} \)
71 \( 1 + (-1.58 - 1.14i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.669 + 0.743i)T^{2} \)
83 \( 1 + (-0.978 + 0.207i)T^{2} \)
89 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.604 - 1.35i)T + (-0.669 + 0.743i)T^{2} \)
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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

−8.899427574314230730620377590328, −8.052166554173191956932260836918, −7.55183992537799984517660973079, −6.64787740045778511982978589622, −5.85839485110082608436782228832, −4.78321785075386819516024107151, −4.03181433003404655690238320011, −3.02669709234822661787628426298, −2.15127724959666809917104205533, −0.74274977221506657780787953122, 1.91093708975232210763136129848, 2.97071696042548722584449839814, 3.57239502782228496917300279448, 4.35694875073350528735736485185, 5.08323857063150767083079725731, 6.65705383634932769355512853207, 7.39813624000276342687756385982, 7.892549988349122632948971762771, 8.307432530968321965140564821972, 9.285475712724609350944067040975

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