Properties

Label 2-3150-15.2-c1-0-12
Degree $2$
Conductor $3150$
Sign $0.161 - 0.986i$
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 + 3.41i·11-s + (−2.02 + 2.02i)13-s − 1.00·14-s − 1.00·16-s + (4.37 − 4.37i)17-s − 4.87i·19-s + (−2.41 − 2.41i)22-s + (3.74 + 3.74i)23-s − 2.86i·26-s + (0.707 − 0.707i)28-s + 5.21·29-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 + 1.02i·11-s + (−0.561 + 0.561i)13-s − 0.267·14-s − 0.250·16-s + (1.06 − 1.06i)17-s − 1.11i·19-s + (−0.514 − 0.514i)22-s + (0.780 + 0.780i)23-s − 0.561i·26-s + (0.133 − 0.133i)28-s + 0.968·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.161 - 0.986i$
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: \((2,\ 3150,\ (\ :1/2),\ 0.161 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.381882692\)
\(L(\frac12)\) \(\approx\) \(1.381882692\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41iT - 11T^{2} \)
13 \( 1 + (2.02 - 2.02i)T - 13iT^{2} \)
17 \( 1 + (-4.37 + 4.37i)T - 17iT^{2} \)
19 \( 1 + 4.87iT - 19T^{2} \)
23 \( 1 + (-3.74 - 3.74i)T + 23iT^{2} \)
29 \( 1 - 5.21T + 29T^{2} \)
31 \( 1 + 4.93T + 31T^{2} \)
37 \( 1 + (-6.87 - 6.87i)T + 37iT^{2} \)
41 \( 1 + 8.77iT - 41T^{2} \)
43 \( 1 + (-0.174 + 0.174i)T - 43iT^{2} \)
47 \( 1 + (3.42 - 3.42i)T - 47iT^{2} \)
53 \( 1 + (-6.43 - 6.43i)T + 53iT^{2} \)
59 \( 1 - 2.22T + 59T^{2} \)
61 \( 1 + 15.2T + 61T^{2} \)
67 \( 1 + (3.81 + 3.81i)T + 67iT^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + (-2.51 + 2.51i)T - 73iT^{2} \)
79 \( 1 + 9.98iT - 79T^{2} \)
83 \( 1 + (-2.35 - 2.35i)T + 83iT^{2} \)
89 \( 1 - 4.07T + 89T^{2} \)
97 \( 1 + (-5.64 - 5.64i)T + 97iT^{2} \)
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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

−9.105888007648190874490852920263, −7.942943811421903208049690477072, −7.30539823187791877485412540897, −6.91442774975638227186657398901, −5.85177446810207224688741861903, −4.96120311613511133279418154484, −4.57194791577040989773107557424, −3.10827393343650203989420938924, −2.18108823679994157617920148507, −0.998958831000997447776626346292, 0.63566087153005768704016647823, 1.65035383270839368215730839981, 2.88911616398679639984627737041, 3.55396518073639348193880556878, 4.49337972918497081270687232821, 5.54876912820462971468465031761, 6.19186115715324552916336473773, 7.24725297545357072437235452397, 8.079649077997763104996973209007, 8.322830110110725385528596257868

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