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Properties

Label 2-3150-15.8-c1-0-34
Degree $2$
Conductor $3150$
Sign $-0.391 + 0.920i$
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 + 0.585i·11-s + (−4 − 4i)13-s + 1.00·14-s − 1.00·16-s + (−0.585 − 0.585i)17-s − 2.82i·19-s + (−0.414 + 0.414i)22-s + (−4.82 + 4.82i)23-s − 5.65i·26-s + (0.707 + 0.707i)28-s + 0.828·29-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 + 0.176i·11-s + (−1.10 − 1.10i)13-s + 0.267·14-s − 0.250·16-s + (−0.142 − 0.142i)17-s − 0.648i·19-s + (−0.0883 + 0.0883i)22-s + (−1.00 + 1.00i)23-s − 1.10i·26-s + (0.133 + 0.133i)28-s + 0.153·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.391 + 0.920i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (2843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -0.391 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6123322223\)
\(L(\frac12)\) \(\approx\) \(0.6123322223\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.585iT - 11T^{2} \)
13 \( 1 + (4 + 4i)T + 13iT^{2} \)
17 \( 1 + (0.585 + 0.585i)T + 17iT^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 + (4.82 - 4.82i)T - 23iT^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 - 1.75T + 31T^{2} \)
37 \( 1 + (6.24 - 6.24i)T - 37iT^{2} \)
41 \( 1 + 3.17iT - 41T^{2} \)
43 \( 1 + (6.07 + 6.07i)T + 43iT^{2} \)
47 \( 1 + (9.24 + 9.24i)T + 47iT^{2} \)
53 \( 1 + (-2.58 + 2.58i)T - 53iT^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 + 9.89T + 61T^{2} \)
67 \( 1 + (8.41 - 8.41i)T - 67iT^{2} \)
71 \( 1 - 1.17iT - 71T^{2} \)
73 \( 1 + (7.07 + 7.07i)T + 73iT^{2} \)
79 \( 1 + 5.65iT - 79T^{2} \)
83 \( 1 + (-0.828 + 0.828i)T - 83iT^{2} \)
89 \( 1 - 3.17T + 89T^{2} \)
97 \( 1 + (-4.58 + 4.58i)T - 97iT^{2} \)
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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

−8.275897141276937566688650806880, −7.54876490870773672807114860941, −7.05316505003664333030627414489, −6.15526081538407633142898494522, −5.19420848483835656906849703847, −4.85506224957871488336247804520, −3.74372292793967583728040003755, −2.94703555562049082065346653831, −1.84912247292260337394285096207, −0.14407430777976863492772997338, 1.58988401323574838412331701364, 2.34874342955912613920407212635, 3.33197555375748029013213590785, 4.41852509076874349325921878710, 4.78112257973831550785790208981, 5.89759883913406181838949175106, 6.46979991395662020674358669529, 7.39278875850083342181389429511, 8.225398039033881296857734072802, 8.999669119563229774586159354840

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