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Properties

Label 2-3150-105.59-c1-0-43
Degree $2$
Conductor $3150$
Sign $-0.935 + 0.354i$
Analytic cond. $25.1528$
Root an. cond. $5.01526$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.358 − 2.62i)7-s + 0.999·8-s + (−2.59 − 1.5i)11-s − 2.44·13-s + (−2.44 + i)14-s + (−0.5 − 0.866i)16-s + (5.12 + 2.95i)17-s + (5.12 − 2.95i)19-s + 3i·22-s + (2.12 + 3.67i)23-s + (1.22 + 2.12i)26-s + (2.09 + 1.62i)28-s − 7.24i·29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.135 − 0.990i)7-s + 0.353·8-s + (−0.783 − 0.452i)11-s − 0.679·13-s + (−0.654 + 0.267i)14-s + (−0.125 − 0.216i)16-s + (1.24 + 0.717i)17-s + (1.17 − 0.678i)19-s + 0.639i·22-s + (0.442 + 0.766i)23-s + (0.240 + 0.416i)26-s + (0.395 + 0.306i)28-s − 1.34i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.935 + 0.354i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -0.935 + 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9971181338\)
\(L(\frac12)\) \(\approx\) \(0.9971181338\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.358 + 2.62i)T \)
good11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
17 \( 1 + (-5.12 - 2.95i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.12 + 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.12 - 3.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.24iT - 29T^{2} \)
31 \( 1 + (7.86 + 4.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.210 + 0.121i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + 0.242iT - 43T^{2} \)
47 \( 1 + (5.12 - 2.95i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.62 + 6.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.03 - 6.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.878 + 0.507i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.66 + 5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.75iT - 71T^{2} \)
73 \( 1 + (-0.717 + 1.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.37 + 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.63iT - 83T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−8.234833127675441162789077524117, −7.55185508136405821109247193417, −7.34600889313453639597071753036, −5.94042373043440272690822559502, −5.24596379436587874517004506483, −4.27836680682684524749952551898, −3.45914942624014748654530272389, −2.67182802459757513713878044661, −1.42381403937196745603907718074, −0.37980302078076454295286044726, 1.28141558610562689170863959346, 2.51364843617247239667898348552, 3.33179311236744859505928227049, 4.77913407573857216977377409521, 5.32411883628217648335473301347, 5.81113433686617261009865356083, 7.02446640263090839683550234728, 7.49582240599904201731707524539, 8.155326749511602659256105027257, 9.061191014325594727237089161496

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