Properties

Label 2-3420-1.1-c1-0-11
Degree $2$
Conductor $3420$
Sign $1$
Analytic cond. $27.3088$
Root an. cond. $5.22578$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·7-s + 6·13-s − 2·17-s − 19-s + 2·23-s + 25-s + 2·29-s + 4·31-s + 2·35-s − 10·37-s + 10·41-s + 6·43-s + 6·47-s − 3·49-s − 6·53-s + 4·59-s + 2·61-s + 6·65-s − 2·67-s − 12·71-s − 6·73-s + 8·79-s + 2·83-s − 2·85-s − 2·89-s + 12·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s + 1.66·13-s − 0.485·17-s − 0.229·19-s + 0.417·23-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.338·35-s − 1.64·37-s + 1.56·41-s + 0.914·43-s + 0.875·47-s − 3/7·49-s − 0.824·53-s + 0.520·59-s + 0.256·61-s + 0.744·65-s − 0.244·67-s − 1.42·71-s − 0.702·73-s + 0.900·79-s + 0.219·83-s − 0.216·85-s − 0.211·89-s + 1.25·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3420} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.513441668\)
\(L(\frac12)\) \(\approx\) \(2.513441668\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
19 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.694851999000613735781495288463, −7.975642425701524882217327634210, −7.08407840488825294244459268304, −6.26185163564153338558652482863, −5.68990131122318397230996089050, −4.74182762688480039947768216585, −4.01886771733175904580642463706, −2.99293759266190349659968382475, −1.91670462040002196308989838706, −1.01289246869200414139939669709, 1.01289246869200414139939669709, 1.91670462040002196308989838706, 2.99293759266190349659968382475, 4.01886771733175904580642463706, 4.74182762688480039947768216585, 5.68990131122318397230996089050, 6.26185163564153338558652482863, 7.08407840488825294244459268304, 7.975642425701524882217327634210, 8.694851999000613735781495288463

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