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Properties

Label 2-1700-1700.1239-c0-0-0
Degree $2$
Conductor $1700$
Sign $0.837 + 0.546i$
Analytic cond. $0.848410$
Root an. cond. $0.921092$
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  + (−0.156 + 0.987i)2-s + (−0.951 − 0.309i)4-s + (−0.951 + 0.309i)5-s + (0.453 − 0.891i)8-s + (−0.987 + 0.156i)9-s + (−0.156 − 0.987i)10-s + (−0.734 − 0.533i)13-s + (0.809 + 0.587i)16-s + (0.587 − 0.809i)17-s − i·18-s + 20-s + (0.809 − 0.587i)25-s + (0.642 − 0.642i)26-s + (−0.152 − 1.93i)29-s + (−0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s + (−0.951 − 0.309i)4-s + (−0.951 + 0.309i)5-s + (0.453 − 0.891i)8-s + (−0.987 + 0.156i)9-s + (−0.156 − 0.987i)10-s + (−0.734 − 0.533i)13-s + (0.809 + 0.587i)16-s + (0.587 − 0.809i)17-s − i·18-s + 20-s + (0.809 − 0.587i)25-s + (0.642 − 0.642i)26-s + (−0.152 − 1.93i)29-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.837 + 0.546i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ 0.837 + 0.546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4475136118\)
\(L(\frac12)\) \(\approx\) \(0.4475136118\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.156 - 0.987i)T \)
5 \( 1 + (0.951 - 0.309i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
good3 \( 1 + (0.987 - 0.156i)T^{2} \)
7 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.891 - 0.453i)T^{2} \)
13 \( 1 + (0.734 + 0.533i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.587 + 0.809i)T^{2} \)
23 \( 1 + (-0.891 + 0.453i)T^{2} \)
29 \( 1 + (0.152 + 1.93i)T + (-0.987 + 0.156i)T^{2} \)
31 \( 1 + (-0.156 + 0.987i)T^{2} \)
37 \( 1 + (-1.65 - 0.398i)T + (0.891 + 0.453i)T^{2} \)
41 \( 1 + (0.678 + 1.10i)T + (-0.453 + 0.891i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.951 + 0.309i)T^{2} \)
61 \( 1 + (1.47 - 0.355i)T + (0.891 - 0.453i)T^{2} \)
67 \( 1 + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (-0.987 + 0.156i)T^{2} \)
73 \( 1 + (-1.70 - 1.04i)T + (0.453 + 0.891i)T^{2} \)
79 \( 1 + (-0.156 - 0.987i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + (-0.183 - 0.253i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-1.04 + 0.0819i)T + (0.987 - 0.156i)T^{2} \)
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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

−9.402190789253013735996695668107, −8.209222315077690750705044482081, −8.010096287574154643724402043946, −7.19615383789917107806842717841, −6.33011986735033216411654473308, −5.44107354805166303661383394022, −4.70126971792462970312153583549, −3.66773852836896879621540260379, −2.66301666039660477451592423566, −0.38918764602838744874940965603, 1.36702451386486016297676511654, 2.80571160727100382348653681224, 3.51644639758850623918155467888, 4.50283765212201168111905084458, 5.20532648779213072476315172581, 6.35267410311366254381686768548, 7.65472362103919808691679564798, 8.076665530767715228733437189611, 9.031992587070554457544887103388, 9.427869986604661473746570776095

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