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Properties

Label 2-3420-285.227-c0-0-5
Degree $2$
Conductor $3420$
Sign $0.980 - 0.198i$
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  + (0.965 − 0.258i)5-s + (1.36 + 1.36i)7-s + 0.517i·11-s + (1.22 − 1.22i)17-s − i·19-s + (−1.41 − 1.41i)23-s + (0.866 − 0.499i)25-s + (1.67 + 0.965i)35-s + (−1.36 + 1.36i)43-s + (−0.707 + 0.707i)47-s + 2.73i·49-s + (0.133 + 0.499i)55-s − 61-s + (−0.366 + 0.366i)73-s + (−0.707 + 0.707i)77-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)5-s + (1.36 + 1.36i)7-s + 0.517i·11-s + (1.22 − 1.22i)17-s − i·19-s + (−1.41 − 1.41i)23-s + (0.866 − 0.499i)25-s + (1.67 + 0.965i)35-s + (−1.36 + 1.36i)43-s + (−0.707 + 0.707i)47-s + 2.73i·49-s + (0.133 + 0.499i)55-s − 61-s + (−0.366 + 0.366i)73-s + (−0.707 + 0.707i)77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.980 - 0.198i$
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: \((2,\ 3420,\ (\ :0),\ 0.980 - 0.198i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.803073074\)
\(L(\frac12)\) \(\approx\) \(1.803073074\)
\(L(1)\) 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 \( 1 + (-0.965 + 0.258i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
11 \( 1 - 0.517iT - T^{2} \)
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 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
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 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
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

−8.740337753789015061551448785106, −8.222553219005127955216908784014, −7.43886196418693240145277790995, −6.36830703339696606755391592062, −5.74698225600029847158283550179, −4.85904755133761686284789844247, −4.69656527082655901076525203535, −2.90847774007001235864028728059, −2.27586665743038215817480069804, −1.39724272816058904603397986690, 1.43120017539595897057389906083, 1.80378981486713392837862221013, 3.47591879296655158453848700915, 3.93028720854572083946454798342, 5.12303021909236582736850105967, 5.68143561238023325049458990776, 6.44785867582148709797742688915, 7.46714113574992049054520070819, 7.947577360215880617424453106754, 8.553167582426956671945298910004

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