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Properties

Label 2-3400-3400.1019-c0-0-3
Degree $2$
Conductor $3400$
Sign $0.844 + 0.535i$
Analytic cond. $1.69682$
Root an. cond. $1.30262$
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  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.707 − 0.707i)5-s + 0.907i·7-s + (0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.453 − 0.891i)10-s + (0.280 + 0.863i)14-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + 0.999i·18-s + (0.951 + 0.690i)19-s + (0.156 − 0.987i)20-s + (−0.297 + 0.0966i)23-s − 1.00i·25-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.707 − 0.707i)5-s + 0.907i·7-s + (0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.453 − 0.891i)10-s + (0.280 + 0.863i)14-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + 0.999i·18-s + (0.951 + 0.690i)19-s + (0.156 − 0.987i)20-s + (−0.297 + 0.0966i)23-s − 1.00i·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3400\)    =    \(2^{3} \cdot 5^{2} \cdot 17\)
Sign: $0.844 + 0.535i$
Analytic conductor: \(1.69682\)
Root analytic conductor: \(1.30262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3400} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3400,\ (\ :0),\ 0.844 + 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.639834877\)
\(L(\frac12)\) \(\approx\) \(2.639834877\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
good3 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 - 0.907iT - T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.297 - 0.0966i)T + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (1.14 - 0.831i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (1.59 + 1.16i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-1.69 - 0.550i)T + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (1.44 - 1.04i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.14 + 0.831i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.309 - 0.951i)T^{2} \)
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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

−8.847257771959855030498904074559, −7.82830698775413916767469380133, −7.25872973787633520763987827227, −5.89122110799020351258249407294, −5.65330672627566420513115601511, −5.10877621240444143042367626677, −4.18210869944463901163721674880, −3.04969706513550876376852132868, −2.27214233888015899991266720740, −1.45074968598873491355058947537, 1.46674119081625083380250324453, 2.66687758492573957196678565269, 3.51158087097606233789362690468, 4.02040947399078654926859489546, 5.20992447348031407072102862152, 5.93455861211445863176612758608, 6.41971775166504097881064868383, 7.39542612060825335328767706291, 7.57239153324414158958078842334, 8.900496668766963990665275493927

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