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Properties

Label 2-3150-5.4-c1-0-38
Degree $2$
Conductor $3150$
Sign $-0.447 + 0.894i$
Analytic cond. $25.1528$
Root an. cond. $5.01526$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + i·7-s − i·8-s + 4i·13-s − 14-s + 16-s − 6i·17-s − 2·19-s − 4·26-s − i·28-s − 6·29-s − 4·31-s + i·32-s + 6·34-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s + 1.10i·13-s − 0.267·14-s + 0.250·16-s − 1.45i·17-s − 0.458·19-s − 0.784·26-s − 0.188i·28-s − 1.11·29-s − 0.718·31-s + 0.176i·32-s + 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (2899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - iT \)
good11 \( 1 + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−8.468418519885528637962214431254, −7.51273754679686785494370501679, −7.01611938799738965261352044548, −6.23276218257571699812215904672, −5.42562084050541798874312134224, −4.70132665847868844630653244600, −3.88213362180117654445561148768, −2.78030602779099191550065475545, −1.65780791050104787857755933063, 0, 1.37016469997267057449178089703, 2.32926584336505779308085888758, 3.50466518349662252289564988165, 3.93951040333211005512152868801, 5.07186281438068485638097795717, 5.74668909721397699602972868634, 6.64672011601897313253102181433, 7.61872275281555138758797831825, 8.276213080788217683721601257206

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