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Properties

Label 2-2070-15.8-c1-0-3
Degree $2$
Conductor $2070$
Sign $0.164 - 0.986i$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−1.88 − 1.20i)5-s + (−1.61 + 1.61i)7-s + (0.707 − 0.707i)8-s + (0.483 + 2.18i)10-s − 3.53i·11-s + (−0.800 − 0.800i)13-s + 2.28·14-s − 1.00·16-s + (−3.53 − 3.53i)17-s − 2.63i·19-s + (1.20 − 1.88i)20-s + (−2.49 + 2.49i)22-s + (−0.707 + 0.707i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.843 − 0.537i)5-s + (−0.609 + 0.609i)7-s + (0.250 − 0.250i)8-s + (0.152 + 0.690i)10-s − 1.06i·11-s + (−0.221 − 0.221i)13-s + 0.609·14-s − 0.250·16-s + (−0.857 − 0.857i)17-s − 0.603i·19-s + (0.268 − 0.421i)20-s + (−0.532 + 0.532i)22-s + (−0.147 + 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.164 - 0.986i$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 0.164 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2461991307\)
\(L(\frac12)\) \(\approx\) \(0.2461991307\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (1.88 + 1.20i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (1.61 - 1.61i)T - 7iT^{2} \)
11 \( 1 + 3.53iT - 11T^{2} \)
13 \( 1 + (0.800 + 0.800i)T + 13iT^{2} \)
17 \( 1 + (3.53 + 3.53i)T + 17iT^{2} \)
19 \( 1 + 2.63iT - 19T^{2} \)
29 \( 1 - 6.66T + 29T^{2} \)
31 \( 1 - 4.22T + 31T^{2} \)
37 \( 1 + (4.02 - 4.02i)T - 37iT^{2} \)
41 \( 1 + 0.501iT - 41T^{2} \)
43 \( 1 + (4.11 + 4.11i)T + 43iT^{2} \)
47 \( 1 + (-3.72 - 3.72i)T + 47iT^{2} \)
53 \( 1 + (-2.30 + 2.30i)T - 53iT^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 + 13.2T + 61T^{2} \)
67 \( 1 + (5.47 - 5.47i)T - 67iT^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + (-9.57 - 9.57i)T + 73iT^{2} \)
79 \( 1 - 0.108iT - 79T^{2} \)
83 \( 1 + (5.67 - 5.67i)T - 83iT^{2} \)
89 \( 1 - 3.06T + 89T^{2} \)
97 \( 1 + (7.17 - 7.17i)T - 97iT^{2} \)
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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

−9.081700146126117661297731760185, −8.706175225936613711967008132335, −8.000245031173796931954114841015, −7.06852450695678252401725964954, −6.26109411916743768507338006701, −5.14771037894840954891933462547, −4.34676597919518496740980534862, −3.21724790614863949428219708697, −2.63982005539356511024780112510, −0.974021941116558616487650723813, 0.12734739065904289125883995199, 1.80691661277028107596804475486, 3.09536096175188035611797961524, 4.16659111630442789311588106978, 4.73447042517511397778971313819, 6.23043735602104316306224710517, 6.65899803019056861409944714631, 7.41837207537759061142347201225, 8.012472102006662005553621129975, 8.860473035858321489551745357347

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