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Properties

Label 2-3325-3325.2203-c0-0-1
Degree $2$
Conductor $3325$
Sign $0.820 + 0.571i$
Analytic cond. $1.65939$
Root an. cond. $1.28817$
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  + (0.994 + 0.104i)4-s + (0.669 − 0.743i)5-s + (0.207 − 0.978i)7-s + (−0.207 + 0.978i)9-s + (0.406 − 0.0864i)11-s + (0.978 + 0.207i)16-s + (−0.707 − 1.84i)17-s + (0.104 + 0.994i)19-s + (0.743 − 0.669i)20-s + (−0.0570 + 1.08i)23-s + (−0.104 − 0.994i)25-s + (0.309 − 0.951i)28-s + (−0.587 − 0.809i)35-s + (−0.309 + 0.951i)36-s + (0.506 + 0.506i)43-s + (0.413 − 0.0434i)44-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)4-s + (0.669 − 0.743i)5-s + (0.207 − 0.978i)7-s + (−0.207 + 0.978i)9-s + (0.406 − 0.0864i)11-s + (0.978 + 0.207i)16-s + (−0.707 − 1.84i)17-s + (0.104 + 0.994i)19-s + (0.743 − 0.669i)20-s + (−0.0570 + 1.08i)23-s + (−0.104 − 0.994i)25-s + (0.309 − 0.951i)28-s + (−0.587 − 0.809i)35-s + (−0.309 + 0.951i)36-s + (0.506 + 0.506i)43-s + (0.413 − 0.0434i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3325\)    =    \(5^{2} \cdot 7 \cdot 19\)
Sign: $0.820 + 0.571i$
Analytic conductor: \(1.65939\)
Root analytic conductor: \(1.28817\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3325} (2203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3325,\ (\ :0),\ 0.820 + 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.864507353\)
\(L(\frac12)\) \(\approx\) \(1.864507353\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.669 + 0.743i)T \)
7 \( 1 + (-0.207 + 0.978i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
good2 \( 1 + (-0.994 - 0.104i)T^{2} \)
3 \( 1 + (0.207 - 0.978i)T^{2} \)
11 \( 1 + (-0.406 + 0.0864i)T + (0.913 - 0.406i)T^{2} \)
13 \( 1 + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.707 + 1.84i)T + (-0.743 + 0.669i)T^{2} \)
23 \( 1 + (0.0570 - 1.08i)T + (-0.994 - 0.104i)T^{2} \)
29 \( 1 + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.669 + 0.743i)T^{2} \)
37 \( 1 + (-0.406 + 0.913i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.506 - 0.506i)T + iT^{2} \)
47 \( 1 + (0.483 + 0.185i)T + (0.743 + 0.669i)T^{2} \)
53 \( 1 + (-0.207 + 0.978i)T^{2} \)
59 \( 1 + (0.104 + 0.994i)T^{2} \)
61 \( 1 + (-0.155 + 0.139i)T + (0.104 - 0.994i)T^{2} \)
67 \( 1 + (-0.743 + 0.669i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (1.05 - 1.62i)T + (-0.406 - 0.913i)T^{2} \)
79 \( 1 + (0.669 - 0.743i)T^{2} \)
83 \( 1 + (-0.292 + 1.84i)T + (-0.951 - 0.309i)T^{2} \)
89 \( 1 + (0.104 - 0.994i)T^{2} \)
97 \( 1 + (0.951 - 0.309i)T^{2} \)
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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

−8.666818557615318248655816396314, −7.81593387734618270806897747472, −7.34602706407447973632994682350, −6.56228869820762778477368882634, −5.66918622687001008916815926831, −5.01559017831194784468353821344, −4.14540631657143257495805179679, −3.01908414713228493377625915886, −2.05289961365812749654320010455, −1.22772078154236624412247996651, 1.57842804403196916334817326786, 2.38450901328866264375405420679, 3.07679365015009383249274774324, 4.09950052507724148128853410355, 5.37316730913468833581165319910, 6.23177324037234106500334415117, 6.38273523286473512894963329934, 7.12407262771524254481103220044, 8.238186960548405771521456175096, 8.896010082808209530223669998504

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