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Properties

Label 2-3150-21.5-c1-0-1
Degree $2$
Conductor $3150$
Sign $-0.997 + 0.0770i$
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 + (−0.295 − 2.62i)7-s + 0.999i·8-s + (−0.570 + 0.329i)11-s + 6.13i·13-s + (1.05 − 2.42i)14-s + (−0.5 + 0.866i)16-s + (−2.43 − 4.22i)17-s + (−6.30 − 3.63i)19-s − 0.659·22-s + (−3.98 − 2.29i)23-s + (−3.06 + 5.31i)26-s + (2.12 − 1.57i)28-s + 8.09i·29-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.111 − 0.993i)7-s + 0.353i·8-s + (−0.172 + 0.0993i)11-s + 1.70i·13-s + (0.282 − 0.648i)14-s + (−0.125 + 0.216i)16-s + (−0.591 − 1.02i)17-s + (−1.44 − 0.834i)19-s − 0.140·22-s + (−0.830 − 0.479i)23-s + (−0.601 + 1.04i)26-s + (0.402 − 0.296i)28-s + 1.50i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.997 + 0.0770i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -0.997 + 0.0770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4268494556\)
\(L(\frac12)\) \(\approx\) \(0.4268494556\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.295 + 2.62i)T \)
good11 \( 1 + (0.570 - 0.329i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.13iT - 13T^{2} \)
17 \( 1 + (2.43 + 4.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.30 + 3.63i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.98 + 2.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.09iT - 29T^{2} \)
31 \( 1 + (-0.759 + 0.438i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.05 - 8.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.25T + 41T^{2} \)
43 \( 1 + 9.03T + 43T^{2} \)
47 \( 1 + (6.00 - 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.5 + 6.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.06 + 7.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0618 + 0.0357i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.666 + 1.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.60iT - 71T^{2} \)
73 \( 1 + (2.44 - 1.41i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.88 - 5.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.44T + 83T^{2} \)
89 \( 1 + (-2.66 + 4.61i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.4iT - 97T^{2} \)
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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

−8.923280914554081845716405123607, −8.304141191008703165006237868851, −7.24638479589134406138370944101, −6.73210556203100903110384389271, −6.36644708102963304482547213071, −4.89592625487516828904195554351, −4.57259074420352320391899188166, −3.78255674040632228824569730753, −2.69462950058557415519985492727, −1.66393858436422392698673956814, 0.094475653938951892950804924922, 1.85049687518661257144683085883, 2.52530278512396953549639681995, 3.56812767821076220240564918983, 4.25104168250149792897821934156, 5.43170881401605845147023612386, 5.84103669325253858493345612115, 6.44425115878494637636221493414, 7.69361714471615084625729181431, 8.325216072887096365601816161219

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