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Properties

Label 2-3420-95.37-c1-0-3
Degree $2$
Conductor $3420$
Sign $-0.845 - 0.533i$
Analytic cond. $27.3088$
Root an. cond. $5.22578$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.17 − 0.539i)5-s + (2.17 + 2.17i)7-s + 1.70·11-s + (−4.43 − 4.43i)13-s + (1.63 + 1.63i)17-s + (−2.80 − 3.34i)19-s + (−3.24 + 3.24i)23-s + (4.41 + 2.34i)25-s + 3.27·29-s + 7.11i·31-s + (−3.53 − 5.87i)35-s + (0.128 − 0.128i)37-s − 3.83i·41-s + (−5.24 + 5.24i)43-s + (−0.908 − 0.908i)47-s + ⋯
L(s)  = 1  + (−0.970 − 0.241i)5-s + (0.820 + 0.820i)7-s + 0.515·11-s + (−1.23 − 1.23i)13-s + (0.395 + 0.395i)17-s + (−0.642 − 0.766i)19-s + (−0.677 + 0.677i)23-s + (0.883 + 0.468i)25-s + 0.608·29-s + 1.27i·31-s + (−0.598 − 0.993i)35-s + (0.0211 − 0.0211i)37-s − 0.598i·41-s + (−0.800 + 0.800i)43-s + (−0.132 − 0.132i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.845 - 0.533i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ -0.845 - 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4453057674\)
\(L(\frac12)\) \(\approx\) \(0.4453057674\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 + 0.539i)T \)
19 \( 1 + (2.80 + 3.34i)T \)
good7 \( 1 + (-2.17 - 2.17i)T + 7iT^{2} \)
11 \( 1 - 1.70T + 11T^{2} \)
13 \( 1 + (4.43 + 4.43i)T + 13iT^{2} \)
17 \( 1 + (-1.63 - 1.63i)T + 17iT^{2} \)
23 \( 1 + (3.24 - 3.24i)T - 23iT^{2} \)
29 \( 1 - 3.27T + 29T^{2} \)
31 \( 1 - 7.11iT - 31T^{2} \)
37 \( 1 + (-0.128 + 0.128i)T - 37iT^{2} \)
41 \( 1 + 3.83iT - 41T^{2} \)
43 \( 1 + (5.24 - 5.24i)T - 43iT^{2} \)
47 \( 1 + (0.908 + 0.908i)T + 47iT^{2} \)
53 \( 1 + (-5.47 - 5.47i)T + 53iT^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 - 2.63T + 61T^{2} \)
67 \( 1 + (-7.71 + 7.71i)T - 67iT^{2} \)
71 \( 1 - 1.51iT - 71T^{2} \)
73 \( 1 + (8.70 - 8.70i)T - 73iT^{2} \)
79 \( 1 + 4.78T + 79T^{2} \)
83 \( 1 + (6.46 - 6.46i)T - 83iT^{2} \)
89 \( 1 + 5.34T + 89T^{2} \)
97 \( 1 + (10.2 - 10.2i)T - 97iT^{2} \)
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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

−8.620917783234451350033644120274, −8.263718007530772622082917402503, −7.54741773152548558021923914024, −6.83204126635048197917498903514, −5.71738294661480842068358303362, −5.04823252551231319910986407339, −4.42790970238661287743527067949, −3.37000567131898561615645050493, −2.52661543452200702694700959073, −1.30532867775978021906462001611, 0.13911864996988207275945532215, 1.54493489543497636231891719722, 2.59986646481807033023809144329, 3.88320707780006349116281428819, 4.31030618367118358264905920233, 4.95515362175223700306808407318, 6.25445753528888767283982597518, 6.94872809584379862534995611306, 7.59228264189417012158494217030, 8.121119915930576338918844685420

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