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Properties

Label 2-2070-15.2-c1-0-12
Degree $2$
Conductor $2070$
Sign $-0.767 - 0.640i$
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 + (1.80 + 1.32i)5-s + (0.949 + 0.949i)7-s + (0.707 + 0.707i)8-s + (−2.21 + 0.339i)10-s + 2.64i·11-s + (2 − 2i)13-s − 1.34·14-s − 1.00·16-s + (−4.94 + 4.94i)17-s + 2.42i·19-s + (1.32 − 1.80i)20-s + (−1.87 − 1.87i)22-s + (−0.707 − 0.707i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.806 + 0.591i)5-s + (0.358 + 0.358i)7-s + (0.250 + 0.250i)8-s + (−0.698 + 0.107i)10-s + 0.797i·11-s + (0.554 − 0.554i)13-s − 0.358·14-s − 0.250·16-s + (−1.20 + 1.20i)17-s + 0.555i·19-s + (0.295 − 0.403i)20-s + (−0.398 − 0.398i)22-s + (−0.147 − 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.767 - 0.640i$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ -0.767 - 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.299136423\)
\(L(\frac12)\) \(\approx\) \(1.299136423\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 - 1.32i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (-0.949 - 0.949i)T + 7iT^{2} \)
11 \( 1 - 2.64iT - 11T^{2} \)
13 \( 1 + (-2 + 2i)T - 13iT^{2} \)
17 \( 1 + (4.94 - 4.94i)T - 17iT^{2} \)
19 \( 1 - 2.42iT - 19T^{2} \)
29 \( 1 - 0.383T + 29T^{2} \)
31 \( 1 + 6.99T + 31T^{2} \)
37 \( 1 + (0.820 + 0.820i)T + 37iT^{2} \)
41 \( 1 - 5.41iT - 41T^{2} \)
43 \( 1 + (2.82 - 2.82i)T - 43iT^{2} \)
47 \( 1 + (-2.19 + 2.19i)T - 47iT^{2} \)
53 \( 1 + (-7.11 - 7.11i)T + 53iT^{2} \)
59 \( 1 - 1.93T + 59T^{2} \)
61 \( 1 + 8.31T + 61T^{2} \)
67 \( 1 + (2.59 + 2.59i)T + 67iT^{2} \)
71 \( 1 - 8.76iT - 71T^{2} \)
73 \( 1 + (-8.26 + 8.26i)T - 73iT^{2} \)
79 \( 1 + 12.2iT - 79T^{2} \)
83 \( 1 + (-5.56 - 5.56i)T + 83iT^{2} \)
89 \( 1 + 5.34T + 89T^{2} \)
97 \( 1 + (-4.46 - 4.46i)T + 97iT^{2} \)
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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

−9.298308415417256246853045371332, −8.683776898617694448442663350883, −7.893762190946930257366619233931, −7.05797976084441760477351702284, −6.26948372431490397371373517456, −5.73651907437867441223662373054, −4.78245782024218464179263633282, −3.65575055394421994091158824934, −2.29557023260309038503630505848, −1.58595750528740029198797640292, 0.53144510573403538970634572867, 1.68746514217652089013130296341, 2.60585442732063146670067085952, 3.82032453096143371376837921413, 4.72400926361669850399680873022, 5.55933965028327472871422840829, 6.57469375448818320777596757339, 7.28902937798543740801193216945, 8.366452842460294903756729424737, 8.993797208039435558559806028914

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