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Properties

Label 2-3150-21.17-c1-0-48
Degree $2$
Conductor $3150$
Sign $-0.729 + 0.684i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (1.29 − 2.30i)7-s − 0.999i·8-s + (1.11 + 0.641i)11-s − 6.14i·13-s + (−0.0298 − 2.64i)14-s + (−0.5 − 0.866i)16-s + (3.26 − 5.64i)17-s + (−5.22 + 3.01i)19-s + 1.28·22-s + (−2.49 + 1.43i)23-s + (−3.07 − 5.32i)26-s + (−1.34 − 2.27i)28-s + 1.35i·29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.490 − 0.871i)7-s − 0.353i·8-s + (0.335 + 0.193i)11-s − 1.70i·13-s + (−0.00798 − 0.707i)14-s + (−0.125 − 0.216i)16-s + (0.790 − 1.37i)17-s + (−1.19 + 0.692i)19-s + 0.273·22-s + (−0.519 + 0.300i)23-s + (−0.602 − 1.04i)26-s + (−0.254 − 0.430i)28-s + 0.250i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.729 + 0.684i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -0.729 + 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.381034518\)
\(L(\frac12)\) \(\approx\) \(2.381034518\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-1.29 + 2.30i)T \)
good11 \( 1 + (-1.11 - 0.641i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.14iT - 13T^{2} \)
17 \( 1 + (-3.26 + 5.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.22 - 3.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.49 - 1.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.35iT - 29T^{2} \)
31 \( 1 + (7.49 + 4.32i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.76 - 8.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.71T + 41T^{2} \)
43 \( 1 + 5.35T + 43T^{2} \)
47 \( 1 + (-0.403 - 0.698i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.77 + 3.33i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.798 + 1.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.50 - 3.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.69 - 4.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + (-10.7 - 6.20i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.59 - 9.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.74T + 83T^{2} \)
89 \( 1 + (1.81 + 3.15i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.76iT - 97T^{2} \)
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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

−8.057421013687131199151728608702, −7.78434144593459265119086602744, −6.87275343813319663675640652001, −5.94201817514295022415454289354, −5.25859882636262083070676171630, −4.46005466033195212876415535959, −3.65188188899768339814256491631, −2.86589223009908667134148019109, −1.66999972345748754845571306153, −0.56932397165040910442995847644, 1.70746580793223182734354475692, 2.36511563293239586738711800548, 3.71210085695875627100706060365, 4.29135464207215267502162790919, 5.13476047570412580357103795954, 6.11949607686388236135910807420, 6.38947091449975943676789644554, 7.43928262312082513283861271099, 8.189183468882877499486877401826, 8.955922656166479077893326488821

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