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Properties

Label 2-3400-3400.1291-c0-0-1
Degree $2$
Conductor $3400$
Sign $-0.728 - 0.684i$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + i·5-s + 1.17·7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.587 + 0.809i)10-s + (0.951 + 0.690i)14-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s − 18-s + (−0.190 + 0.587i)19-s + (−0.951 + 0.309i)20-s + (−1.53 − 1.11i)23-s − 25-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + i·5-s + 1.17·7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.587 + 0.809i)10-s + (0.951 + 0.690i)14-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s − 18-s + (−0.190 + 0.587i)19-s + (−0.951 + 0.309i)20-s + (−1.53 − 1.11i)23-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3400\)    =    \(2^{3} \cdot 5^{2} \cdot 17\)
Sign: $-0.728 - 0.684i$
Analytic conductor: \(1.69682\)
Root analytic conductor: \(1.30262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3400} (1291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3400,\ (\ :0),\ -0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.986441407\)
\(L(\frac12)\) \(\approx\) \(1.986441407\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 - iT \)
17 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 - 1.17T + T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−8.596314486550745062110515980910, −8.048445501800206708348139713493, −7.73207934262638915365086500085, −6.70026766614688469567659814454, −5.89148851657884067220992613776, −5.58201476514184375109311101371, −4.19512829338765510824628219280, −4.05418982213634764719452374132, −2.47423011439220534155384882625, −2.23621479430147825065564347175, 0.890272881467871053532879402573, 1.91389070507073600389533484606, 2.88827597106357073063626652720, 3.98300156963000742999688165917, 4.65934430828019179033694551683, 5.31151053747254953813572247908, 5.89944591251046486455153142799, 6.87657501681049660135609889939, 7.88076567318473698482694670264, 8.584280968114186862102447422921

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