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Properties

Label 2-3325-3325.2146-c0-0-1
Degree $2$
Conductor $3325$
Sign $0.855 + 0.518i$
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  + (0.913 − 0.406i)4-s + (−0.978 − 0.207i)5-s + (0.669 − 0.743i)7-s + (0.669 − 0.743i)9-s + (0.895 + 0.994i)11-s + (0.669 − 0.743i)16-s + (0.169 + 1.60i)17-s + (0.913 + 0.406i)19-s + (−0.978 + 0.207i)20-s + (−1.30 + 0.278i)23-s + (0.913 + 0.406i)25-s + (0.309 − 0.951i)28-s + (−0.809 + 0.587i)35-s + (0.309 − 0.951i)36-s − 0.209·43-s + (1.22 + 0.544i)44-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)4-s + (−0.978 − 0.207i)5-s + (0.669 − 0.743i)7-s + (0.669 − 0.743i)9-s + (0.895 + 0.994i)11-s + (0.669 − 0.743i)16-s + (0.169 + 1.60i)17-s + (0.913 + 0.406i)19-s + (−0.978 + 0.207i)20-s + (−1.30 + 0.278i)23-s + (0.913 + 0.406i)25-s + (0.309 − 0.951i)28-s + (−0.809 + 0.587i)35-s + (0.309 − 0.951i)36-s − 0.209·43-s + (1.22 + 0.544i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3325\)    =    \(5^{2} \cdot 7 \cdot 19\)
Sign: $0.855 + 0.518i$
Analytic conductor: \(1.65939\)
Root analytic conductor: \(1.28817\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3325} (2146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3325,\ (\ :0),\ 0.855 + 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.629896271\)
\(L(\frac12)\) \(\approx\) \(1.629896271\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.978 + 0.207i)T \)
7 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
good2 \( 1 + (-0.913 + 0.406i)T^{2} \)
3 \( 1 + (-0.669 + 0.743i)T^{2} \)
11 \( 1 + (-0.895 - 0.994i)T + (-0.104 + 0.994i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.169 - 1.60i)T + (-0.978 + 0.207i)T^{2} \)
23 \( 1 + (1.30 - 0.278i)T + (0.913 - 0.406i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.978 - 0.207i)T^{2} \)
37 \( 1 + (0.104 + 0.994i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + 0.209T + T^{2} \)
47 \( 1 + (-0.104 + 0.994i)T + (-0.978 - 0.207i)T^{2} \)
53 \( 1 + (-0.669 + 0.743i)T^{2} \)
59 \( 1 + (-0.913 - 0.406i)T^{2} \)
61 \( 1 + (1.78 - 0.379i)T + (0.913 - 0.406i)T^{2} \)
67 \( 1 + (0.978 - 0.207i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \)
79 \( 1 + (0.978 + 0.207i)T^{2} \)
83 \( 1 + (-0.169 + 0.122i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (-0.913 + 0.406i)T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−8.610777167097304100879379109450, −7.71084465074251087319628289517, −7.38211721378098410446270962136, −6.62163066494289993028403407573, −5.89702246268038056214690082313, −4.73445043220888966046302341340, −3.97666538918420447289370935384, −3.49854292253451760257430864916, −1.78993117979823484958525506183, −1.25982092506756185889585978157, 1.32018958472234252138875192984, 2.51934274417005110749127923333, 3.20659155832287738534148572204, 4.17918856841920125188924443629, 5.00558883807286364287187600616, 5.95345701476326104619369458372, 6.78778599851266975713075814638, 7.56241661626538331040851831310, 7.895265751030419064746216233677, 8.697566154414844744077382287568

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