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Properties

Label 2-3420-285.227-c0-0-4
Degree $2$
Conductor $3420$
Sign $0.787 + 0.615i$
Analytic cond. $1.70680$
Root an. cond. $1.30644$
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  + (0.258 − 0.965i)5-s + (−0.366 − 0.366i)7-s + 1.93i·11-s + (1.22 − 1.22i)17-s − i·19-s + (1.41 + 1.41i)23-s + (−0.866 − 0.499i)25-s + (−0.448 + 0.258i)35-s + (0.366 − 0.366i)43-s + (0.707 − 0.707i)47-s − 0.732i·49-s + (1.86 + 0.499i)55-s − 61-s + (1.36 − 1.36i)73-s + (0.707 − 0.707i)77-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)5-s + (−0.366 − 0.366i)7-s + 1.93i·11-s + (1.22 − 1.22i)17-s − i·19-s + (1.41 + 1.41i)23-s + (−0.866 − 0.499i)25-s + (−0.448 + 0.258i)35-s + (0.366 − 0.366i)43-s + (0.707 − 0.707i)47-s − 0.732i·49-s + (1.86 + 0.499i)55-s − 61-s + (1.36 − 1.36i)73-s + (0.707 − 0.707i)77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.787 + 0.615i$
Analytic conductor: \(1.70680\)
Root analytic conductor: \(1.30644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (3077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :0),\ 0.787 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.333946999\)
\(L(\frac12)\) \(\approx\) \(1.333946999\)
\(L(1)\) 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 \( 1 + (-0.258 + 0.965i)T \)
19 \( 1 + iT \)
good7 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
11 \( 1 - 1.93iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
23 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
47 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.998514876762376102713684645654, −7.72130160977110981066134188606, −7.32765765562538489428911557396, −6.64794669856269170416453635743, −5.29809054011287065828376594316, −5.08034533213587497711196158494, −4.19421566396788458193642454659, −3.14845991911393060235128940043, −2.04776090416592844066342117014, −0.962978049269753485387334396222, 1.20215037093071346453390279227, 2.65703597581308873831471597091, 3.23021733854152714002124830559, 3.94726253803037054607800921498, 5.34257426181552884005561014223, 6.12970515784325624163041679143, 6.25632090678540533035559709314, 7.41472392006847521734580880011, 8.188639209592098344289279799612, 8.746335830200832539294586696183

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