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Properties

Label 2-3150-105.59-c1-0-9
Degree $2$
Conductor $3150$
Sign $-0.961 - 0.274i$
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 + (2.61 − 0.397i)7-s − 0.999·8-s + (−0.429 − 0.248i)11-s − 2.74·13-s + (1.65 + 2.06i)14-s + (−0.5 − 0.866i)16-s + (−3.16 − 1.82i)17-s + (−3.12 + 1.80i)19-s − 0.496i·22-s + (3.21 + 5.56i)23-s + (−1.37 − 2.37i)26-s + (−0.963 + 2.46i)28-s + 8.87i·29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.988 − 0.150i)7-s − 0.353·8-s + (−0.129 − 0.0748i)11-s − 0.761·13-s + (0.441 + 0.552i)14-s + (−0.125 − 0.216i)16-s + (−0.768 − 0.443i)17-s + (−0.716 + 0.413i)19-s − 0.105i·22-s + (0.669 + 1.16i)23-s + (−0.269 − 0.466i)26-s + (−0.182 + 0.465i)28-s + 1.64i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.961 - 0.274i$
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: \((2,\ 3150,\ (\ :1/2),\ -0.961 - 0.274i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.260765390\)
\(L(\frac12)\) \(\approx\) \(1.260765390\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.61 + 0.397i)T \)
good11 \( 1 + (0.429 + 0.248i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.74T + 13T^{2} \)
17 \( 1 + (3.16 + 1.82i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.12 - 1.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.21 - 5.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.87iT - 29T^{2} \)
31 \( 1 + (6.90 + 3.98i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.98 - 1.14i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.22T + 41T^{2} \)
43 \( 1 - 2.22iT - 43T^{2} \)
47 \( 1 + (5.66 - 3.27i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.88 - 6.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.05 - 5.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.24 + 1.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.08 - 4.08i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (6.53 - 11.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.44 + 7.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.79iT - 83T^{2} \)
89 \( 1 + (-0.743 - 1.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.05T + 97T^{2} \)
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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

−8.891986992213174075135019809220, −8.165410984530668635020868340187, −7.35745506056837132893921649311, −7.00244520195917640089509616169, −5.87578306676089418777881300621, −5.16491339159042287893194016900, −4.58425333512683767312643707888, −3.68945439375628911058197508479, −2.59123039281719833379420500394, −1.48003482816701156086523002195, 0.32410038939864024302198807170, 1.88820930338494103894212578508, 2.38747459798087473732937105783, 3.59491201983039454681773274069, 4.64407188244497447429048608129, 4.88147260083685766574165069243, 5.96951459548547351863747068230, 6.77687867638837961523809184164, 7.65658562839446383102540685888, 8.507264005239385496187202134891

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