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Properties

Label 2-5070-13.12-c1-0-89
Degree $2$
Conductor $5070$
Sign $-0.832 - 0.554i$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + 3-s − 4-s − i·5-s − i·6-s + 4i·7-s + i·8-s + 9-s − 10-s − 12-s + 4·14-s − i·15-s + 16-s − 6·17-s − i·18-s − 4i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s + 1.51i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.288·12-s + 1.06·14-s − 0.258i·15-s + 0.250·16-s − 1.45·17-s − 0.235i·18-s − 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(40.4841\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5070} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - T \)
5 \( 1 + iT \)
13 \( 1 \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 8iT - 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 18iT - 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.127556002279082142514529988207, −7.16198049135809868549882999993, −6.30364501080472299132614678930, −5.39791813652783019594829251342, −4.81585939898988208509292805131, −3.97252259699027145899570373537, −2.96959076936302955068544631647, −2.33400265261670445054979910362, −1.60991187752735837712633891342, 0, 1.40311373771204694994124485920, 2.54550686686188345017479747572, 3.71790943787385589931687958200, 4.08369078714151938323789688344, 4.87711270690896588078666881478, 6.06106737387786767843239252016, 6.58050436458872900406455748756, 7.38060663391904745492757492819, 7.73562877633696931316770037680

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