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Properties

Label 2-1700-1700.1467-c0-0-0
Degree $2$
Conductor $1700$
Sign $-0.0231 + 0.999i$
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  + (−0.996 − 0.0784i)2-s + (0.987 + 0.156i)4-s + (0.233 − 0.972i)5-s + (−0.972 − 0.233i)8-s + (0.649 − 0.760i)9-s + (−0.309 + 0.951i)10-s + (0.322 − 0.993i)13-s + (0.951 + 0.309i)16-s + (−0.987 − 0.156i)17-s + (−0.707 + 0.707i)18-s + (0.382 − 0.923i)20-s + (−0.891 − 0.453i)25-s + (−0.399 + 0.965i)26-s + (0.0984 + 0.213i)29-s + (−0.923 − 0.382i)32-s + ⋯
L(s)  = 1  + (−0.996 − 0.0784i)2-s + (0.987 + 0.156i)4-s + (0.233 − 0.972i)5-s + (−0.972 − 0.233i)8-s + (0.649 − 0.760i)9-s + (−0.309 + 0.951i)10-s + (0.322 − 0.993i)13-s + (0.951 + 0.309i)16-s + (−0.987 − 0.156i)17-s + (−0.707 + 0.707i)18-s + (0.382 − 0.923i)20-s + (−0.891 − 0.453i)25-s + (−0.399 + 0.965i)26-s + (0.0984 + 0.213i)29-s + (−0.923 − 0.382i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0231 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0231 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $-0.0231 + 0.999i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ -0.0231 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7346316130\)
\(L(\frac12)\) \(\approx\) \(0.7346316130\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.996 + 0.0784i)T \)
5 \( 1 + (-0.233 + 0.972i)T \)
17 \( 1 + (0.987 + 0.156i)T \)
good3 \( 1 + (-0.649 + 0.760i)T^{2} \)
7 \( 1 + (0.382 + 0.923i)T^{2} \)
11 \( 1 + (0.852 + 0.522i)T^{2} \)
13 \( 1 + (-0.322 + 0.993i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.891 - 0.453i)T^{2} \)
23 \( 1 + (-0.522 + 0.852i)T^{2} \)
29 \( 1 + (-0.0984 - 0.213i)T + (-0.649 + 0.760i)T^{2} \)
31 \( 1 + (-0.996 - 0.0784i)T^{2} \)
37 \( 1 + (-0.473 - 0.265i)T + (0.522 + 0.852i)T^{2} \)
41 \( 1 + (0.831 + 0.0984i)T + (0.972 + 0.233i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.453 - 0.108i)T + (0.891 - 0.453i)T^{2} \)
59 \( 1 + (-0.156 - 0.987i)T^{2} \)
61 \( 1 + (-0.916 - 1.63i)T + (-0.522 + 0.852i)T^{2} \)
67 \( 1 + (-0.951 + 0.309i)T^{2} \)
71 \( 1 + (-0.760 - 0.649i)T^{2} \)
73 \( 1 + (-0.0486 + 0.0616i)T + (-0.233 - 0.972i)T^{2} \)
79 \( 1 + (0.996 - 0.0784i)T^{2} \)
83 \( 1 + (-0.453 + 0.891i)T^{2} \)
89 \( 1 + (-0.690 + 1.35i)T + (-0.587 - 0.809i)T^{2} \)
97 \( 1 + (-1.63 - 0.603i)T + (0.760 + 0.649i)T^{2} \)
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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

−9.256910990375064246238947225754, −8.653251332635037685192634490648, −7.992854057391472718331419695804, −7.03807783951414100123849514344, −6.28493641790589486322975363989, −5.39916530846693251942847457135, −4.27837085611380528909269129353, −3.19228968529958719625287581219, −1.89601992062369554032376323186, −0.790077141867558966896586057069, 1.72991124010168544348519011316, 2.43412002997145877748232174260, 3.67764352335201610649192486485, 4.85175908585683672975056477803, 6.14329901629636612452447311143, 6.66805970225962176343153532106, 7.37439763746551409002586795325, 8.119442327443998737559756807442, 9.029132275987728047619814637390, 9.751590460515881259123362486492

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