Properties

Label 2-5070-13.12-c1-0-2
Degree $2$
Conductor $5070$
Sign $-0.999 - 0.0304i$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
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  + i·2-s − 3-s − 4-s i·5-s i·6-s + 1.19i·7-s i·8-s + 9-s + 10-s − 3.93i·11-s + 12-s − 1.19·14-s + i·15-s + 16-s − 1.74·17-s + i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s + 0.452i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s − 1.18i·11-s + 0.288·12-s − 0.320·14-s + 0.258i·15-s + 0.250·16-s − 0.422·17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.999 - 0.0304i$
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: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ -0.999 - 0.0304i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2642260137\)
\(L(\frac12)\) \(\approx\) \(0.2642260137\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.19iT - 7T^{2} \)
11 \( 1 + 3.93iT - 11T^{2} \)
17 \( 1 + 1.74T + 17T^{2} \)
19 \( 1 - 0.911iT - 19T^{2} \)
23 \( 1 + 1.24T + 23T^{2} \)
29 \( 1 + 3.47T + 29T^{2} \)
31 \( 1 + 2.15iT - 31T^{2} \)
37 \( 1 + 4.80iT - 37T^{2} \)
41 \( 1 - 0.198iT - 41T^{2} \)
43 \( 1 + 3.43T + 43T^{2} \)
47 \( 1 + 0.902iT - 47T^{2} \)
53 \( 1 + 6.91T + 53T^{2} \)
59 \( 1 - 10.2iT - 59T^{2} \)
61 \( 1 - 9.92T + 61T^{2} \)
67 \( 1 + 9.16iT - 67T^{2} \)
71 \( 1 - 11.0iT - 71T^{2} \)
73 \( 1 + 4.13iT - 73T^{2} \)
79 \( 1 - 2.73T + 79T^{2} \)
83 \( 1 - 15.5iT - 83T^{2} \)
89 \( 1 + 8.57iT - 89T^{2} \)
97 \( 1 - 16.9iT - 97T^{2} \)
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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

−8.476060441460132339563649033942, −7.930091793254996476324182386941, −7.05687438714761721833710485341, −6.30083423530242615495459567291, −5.67480483297309462400098251448, −5.25242770349810533147357850405, −4.26752923216014626568798276477, −3.56334698325609932847959849054, −2.32534325939617105434455657022, −1.03087951186062140500447256184, 0.088394300558686218319412513190, 1.46210012705759021330957077697, 2.27432716196676775544224912232, 3.31203156791596538353386672196, 4.17991788459464686052686484190, 4.76647880556174834297057033258, 5.56685159382157903939288950518, 6.56320772888183143582877974786, 7.06633058715250098841436370778, 7.81231822606244726999125060242

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