Properties

Label 2-3150-105.59-c1-0-38
Degree $2$
Conductor $3150$
Sign $-0.845 + 0.534i$
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 + (1.04 + 2.43i)7-s + 0.999·8-s + (−1.38 − 0.800i)11-s + 0.770·13-s + (1.58 − 2.11i)14-s + (−0.5 − 0.866i)16-s + (−3.05 − 1.76i)17-s + (−3.06 + 1.77i)19-s + 1.60i·22-s + (−1.61 − 2.79i)23-s + (−0.385 − 0.667i)26-s + (−2.62 − 0.313i)28-s − 0.700i·29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.393 + 0.919i)7-s + 0.353·8-s + (−0.417 − 0.241i)11-s + 0.213·13-s + (0.423 − 0.566i)14-s + (−0.125 − 0.216i)16-s + (−0.739 − 0.427i)17-s + (−0.703 + 0.406i)19-s + 0.341i·22-s + (−0.336 − 0.582i)23-s + (−0.0755 − 0.130i)26-s + (−0.496 − 0.0592i)28-s − 0.130i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.845 + 0.534i$
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: \((2,\ 3150,\ (\ :1/2),\ -0.845 + 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6003702279\)
\(L(\frac12)\) \(\approx\) \(0.6003702279\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.04 - 2.43i)T \)
good11 \( 1 + (1.38 + 0.800i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.770T + 13T^{2} \)
17 \( 1 + (3.05 + 1.76i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.06 - 1.77i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.61 + 2.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.700iT - 29T^{2} \)
31 \( 1 + (-1.13 - 0.656i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.792 - 0.457i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.88T + 41T^{2} \)
43 \( 1 + 9.26iT - 43T^{2} \)
47 \( 1 + (2.31 - 1.33i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.64 + 8.04i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.56 + 2.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.43 - 5.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.90 - 3.40i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.47iT - 71T^{2} \)
73 \( 1 + (-5.51 + 9.55i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.45 - 2.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.9iT - 83T^{2} \)
89 \( 1 + (-4.40 - 7.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.31T + 97T^{2} \)
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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

−8.521403582013638950327015476582, −7.943477032585766252797571514742, −6.90853860722604273813302790776, −6.08982137833170171245170492143, −5.20834804681175228982375523947, −4.45581330673461514882944978102, −3.43767445852650394967839208569, −2.46270616488283405785672583098, −1.79651309781589958005511566535, −0.21699875183714901111904022029, 1.19164492384188109190547396285, 2.29473280049695765252107787609, 3.66685797189467145224298353966, 4.46991741796038936423679022639, 5.13715077510163823445058783229, 6.18998723955454251064244621349, 6.77616393388181354412902430078, 7.56739824105545684603811204378, 8.136187878146096544755571887098, 8.832715144418760949975647269032

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