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Properties

Label 2-3420-15.8-c1-0-7
Degree $2$
Conductor $3420$
Sign $0.118 - 0.992i$
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  + (0.401 − 2.19i)5-s + (−0.0703 + 0.0703i)7-s + 1.51i·11-s + (2.83 + 2.83i)13-s + (5.16 + 5.16i)17-s + i·19-s + (−6.16 + 6.16i)23-s + (−4.67 − 1.76i)25-s − 7.51·29-s − 0.487·31-s + (0.126 + 0.182i)35-s + (3.77 − 3.77i)37-s + 5.91i·41-s + (−1.49 − 1.49i)43-s + (−6.26 − 6.26i)47-s + ⋯
L(s)  = 1  + (0.179 − 0.983i)5-s + (−0.0265 + 0.0265i)7-s + 0.456i·11-s + (0.786 + 0.786i)13-s + (1.25 + 1.25i)17-s + 0.229i·19-s + (−1.28 + 1.28i)23-s + (−0.935 − 0.353i)25-s − 1.39·29-s − 0.0875·31-s + (0.0213 + 0.0309i)35-s + (0.620 − 0.620i)37-s + 0.923i·41-s + (−0.227 − 0.227i)43-s + (−0.913 − 0.913i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.118 - 0.992i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (1673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ 0.118 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.379527265\)
\(L(\frac12)\) \(\approx\) \(1.379527265\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.401 + 2.19i)T \)
19 \( 1 - iT \)
good7 \( 1 + (0.0703 - 0.0703i)T - 7iT^{2} \)
11 \( 1 - 1.51iT - 11T^{2} \)
13 \( 1 + (-2.83 - 2.83i)T + 13iT^{2} \)
17 \( 1 + (-5.16 - 5.16i)T + 17iT^{2} \)
23 \( 1 + (6.16 - 6.16i)T - 23iT^{2} \)
29 \( 1 + 7.51T + 29T^{2} \)
31 \( 1 + 0.487T + 31T^{2} \)
37 \( 1 + (-3.77 + 3.77i)T - 37iT^{2} \)
41 \( 1 - 5.91iT - 41T^{2} \)
43 \( 1 + (1.49 + 1.49i)T + 43iT^{2} \)
47 \( 1 + (6.26 + 6.26i)T + 47iT^{2} \)
53 \( 1 + (3.78 - 3.78i)T - 53iT^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 + 8.31T + 61T^{2} \)
67 \( 1 + (2.19 - 2.19i)T - 67iT^{2} \)
71 \( 1 + 4.13iT - 71T^{2} \)
73 \( 1 + (-6.69 - 6.69i)T + 73iT^{2} \)
79 \( 1 + 1.31iT - 79T^{2} \)
83 \( 1 + (1.02 - 1.02i)T - 83iT^{2} \)
89 \( 1 - 7.02T + 89T^{2} \)
97 \( 1 + (0.755 - 0.755i)T - 97iT^{2} \)
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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

−8.844814816652019413376933430998, −7.86770000510099018312349210525, −7.66091554107947651783717483676, −6.20040051554187117869827086573, −5.90876268790306823068077721288, −4.98929427234088092868693774346, −4.03594162612122633479805340949, −3.53262780903214855928692387969, −1.87761623825810966595622883960, −1.39170345030322235312925481288, 0.40964500648162947074015101662, 1.86212565038234564903215662328, 3.03121462637117324475539591460, 3.41290811703220180564875494583, 4.57158941357230140905303038650, 5.66003736760572045981950270089, 6.08191032905262249616738918044, 6.94255134090492715543242534223, 7.75600949888511651266483735229, 8.216957374132644241843505899500

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