Properties

Label 2-3150-15.2-c1-0-30
Degree $2$
Conductor $3150$
Sign $-0.920 + 0.391i$
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  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s − 4.82i·11-s + (−3.41 + 3.41i)13-s + 1.00·14-s − 1.00·16-s + (1.58 − 1.58i)17-s − 7.07i·19-s + (−3.41 − 3.41i)22-s + (4.82 + 4.82i)23-s + 4.82i·26-s + (0.707 − 0.707i)28-s − 3.17·29-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s − 1.45i·11-s + (−0.946 + 0.946i)13-s + 0.267·14-s − 0.250·16-s + (0.384 − 0.384i)17-s − 1.62i·19-s + (−0.727 − 0.727i)22-s + (1.00 + 1.00i)23-s + 0.946i·26-s + (0.133 − 0.133i)28-s − 0.588·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.920 + 0.391i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -0.920 + 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.532248963\)
\(L(\frac12)\) \(\approx\) \(1.532248963\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 + 4.82iT - 11T^{2} \)
13 \( 1 + (3.41 - 3.41i)T - 13iT^{2} \)
17 \( 1 + (-1.58 + 1.58i)T - 17iT^{2} \)
19 \( 1 + 7.07iT - 19T^{2} \)
23 \( 1 + (-4.82 - 4.82i)T + 23iT^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 + 8.82T + 31T^{2} \)
37 \( 1 + (6.07 + 6.07i)T + 37iT^{2} \)
41 \( 1 - 4.82iT - 41T^{2} \)
43 \( 1 + (-6.41 + 6.41i)T - 43iT^{2} \)
47 \( 1 + (-6.41 + 6.41i)T - 47iT^{2} \)
53 \( 1 + (4.82 + 4.82i)T + 53iT^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 6.58T + 61T^{2} \)
67 \( 1 + (-0.414 - 0.414i)T + 67iT^{2} \)
71 \( 1 - 12.2iT - 71T^{2} \)
73 \( 1 + (6 - 6i)T - 73iT^{2} \)
79 \( 1 + 13.3iT - 79T^{2} \)
83 \( 1 + (9.65 + 9.65i)T + 83iT^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (8.24 + 8.24i)T + 97iT^{2} \)
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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

−8.697538581212803081723856836819, −7.25925375112465015905152001673, −7.07219654368185706993912186260, −5.69457033357114548571134118197, −5.37446865885741896620333997673, −4.44772554609412277626013988481, −3.48593459473262998273581175468, −2.73638249043570679269708135855, −1.73929204026715908590831042123, −0.37819456128522002662235569882, 1.53175132943956846750161837661, 2.61622580916779869568255226057, 3.65284204727890436855739749492, 4.48001283313385336046514333623, 5.18075396463620687253856508172, 5.86788169274077302047597449761, 6.86233358219204620456258069172, 7.57089556821765262214746383896, 7.87234798792932557341452187373, 8.936437740353928077996887872229

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