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Properties

Label 2-3150-105.59-c1-0-13
Degree $2$
Conductor $3150$
Sign $-0.766 - 0.641i$
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 + (−2.63 + 0.189i)7-s − 0.999·8-s + (2.55 + 1.47i)11-s + 3.93·13-s + (−1.48 − 2.19i)14-s + (−0.5 − 0.866i)16-s + (−0.346 − 0.199i)17-s + (−0.0305 + 0.0176i)19-s + 2.94i·22-s + (−1.86 − 3.23i)23-s + (1.96 + 3.40i)26-s + (1.15 − 2.38i)28-s + 8.89i·29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.997 + 0.0716i)7-s − 0.353·8-s + (0.769 + 0.444i)11-s + 1.09·13-s + (−0.396 − 0.585i)14-s + (−0.125 − 0.216i)16-s + (−0.0839 − 0.0484i)17-s + (−0.00700 + 0.00404i)19-s + 0.628i·22-s + (−0.389 − 0.673i)23-s + (0.385 + 0.667i)26-s + (0.218 − 0.449i)28-s + 1.65i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.766 - 0.641i$
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: \((2,\ 3150,\ (\ :1/2),\ -0.766 - 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.573374225\)
\(L(\frac12)\) \(\approx\) \(1.573374225\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 0.189i)T \)
good11 \( 1 + (-2.55 - 1.47i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.93T + 13T^{2} \)
17 \( 1 + (0.346 + 0.199i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0305 - 0.0176i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.86 + 3.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.89iT - 29T^{2} \)
31 \( 1 + (-0.717 - 0.414i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.86 - 3.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.31T + 41T^{2} \)
43 \( 1 - 3.03iT - 43T^{2} \)
47 \( 1 + (-5.02 + 2.90i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.14 - 3.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.78 - 4.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.97 - 5.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.8 - 6.25i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.93iT - 71T^{2} \)
73 \( 1 + (0.171 - 0.297i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.15 - 7.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (-3.08 - 5.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−8.946400084717607259524474062400, −8.261899592935209028071031500007, −7.22477379774655571519839791773, −6.64786850304236924177649889424, −6.11644316168748301677909291489, −5.27113871157056300116760742097, −4.23737662443321291910153181597, −3.61832168351291675971930365111, −2.71588772663555874532421245603, −1.25908288896684533601914407487, 0.46320341011637138737297652369, 1.67267661890187234759616252268, 2.82016139626426568982749585306, 3.74661729727510632603433110249, 4.07448264434652889327833044511, 5.41029455434981975527503904510, 6.13600630772294026379670585738, 6.58455445184218065233871304338, 7.68789592963013355332390242020, 8.566668689327022992290693441329

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