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Properties

Label 2-3150-105.89-c1-0-35
Degree $2$
Conductor $3150$
Sign $-0.228 + 0.973i$
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 + (−0.189 + 2.63i)7-s + 0.999·8-s + (−1.32 + 0.766i)11-s + 1.48·13-s + (−2.19 − 1.48i)14-s + (−0.5 + 0.866i)16-s + (−2.10 + 1.21i)17-s + (−4.21 − 2.43i)19-s − 1.53i·22-s + (0.133 − 0.232i)23-s + (−0.741 + 1.28i)26-s + (2.38 − 1.15i)28-s + 0.898i·29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.0716 + 0.997i)7-s + 0.353·8-s + (−0.400 + 0.230i)11-s + 0.411·13-s + (−0.585 − 0.396i)14-s + (−0.125 + 0.216i)16-s + (−0.510 + 0.294i)17-s + (−0.966 − 0.557i)19-s − 0.326i·22-s + (0.0279 − 0.0483i)23-s + (−0.145 + 0.251i)26-s + (0.449 − 0.218i)28-s + 0.166i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.228 + 0.973i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -0.228 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1117494041\)
\(L(\frac12)\) \(\approx\) \(0.1117494041\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.189 - 2.63i)T \)
good11 \( 1 + (1.32 - 0.766i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.48T + 13T^{2} \)
17 \( 1 + (2.10 - 1.21i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.21 + 2.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.133 + 0.232i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.898iT - 29T^{2} \)
31 \( 1 + (0.717 - 0.414i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.74 - 2.74i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.76T + 41T^{2} \)
43 \( 1 + 1.86iT - 43T^{2} \)
47 \( 1 + (6.46 + 3.72i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.73 + 3.00i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.12 + 5.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.73 - 3.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-13.8 + 8.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + (0.171 + 0.297i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.22 - 9.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.45iT - 83T^{2} \)
89 \( 1 + (7.98 - 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−8.358430036297233182109270899450, −7.986796773250992278629155815432, −6.64595362133289283687947177801, −6.53318190322133333056970479307, −5.40876406744069022131765694558, −4.88609155050153573183539380672, −3.80772882306553630208670865616, −2.63031878760992398581700258274, −1.72939123951337925896424636787, −0.04055034230630305447547354934, 1.18902390648819687698772754582, 2.28397528987748792368155133473, 3.33021264999349982743765308567, 4.08776800517695208003111855258, 4.81299644634773313634708185153, 5.94798133029499561533899968860, 6.75470130973485850695763199032, 7.53361189670578802012503617551, 8.269090128789887184716362978267, 8.838241131208816793807323986980

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