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Properties

Label 2-3150-105.104-c1-0-21
Degree $2$
Conductor $3150$
Sign $0.440 - 0.897i$
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  + 2-s + 4-s + (2.64 − 0.0951i)7-s + 8-s + 5.28i·11-s + 2.19·13-s + (2.64 − 0.0951i)14-s + 16-s + 1.04i·17-s + 6.43i·19-s + 5.28i·22-s − 7.47·23-s + 2.19·26-s + (2.64 − 0.0951i)28-s − 7.47i·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.999 − 0.0359i)7-s + 0.353·8-s + 1.59i·11-s + 0.607·13-s + (0.706 − 0.0254i)14-s + 0.250·16-s + 0.253i·17-s + 1.47i·19-s + 1.12i·22-s − 1.55·23-s + 0.429·26-s + (0.499 − 0.0179i)28-s − 1.38i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.440 - 0.897i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (3149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ 0.440 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.190888935\)
\(L(\frac12)\) \(\approx\) \(3.190888935\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.64 + 0.0951i)T \)
good11 \( 1 - 5.28iT - 11T^{2} \)
13 \( 1 - 2.19T + 13T^{2} \)
17 \( 1 - 1.04iT - 17T^{2} \)
19 \( 1 - 6.43iT - 19T^{2} \)
23 \( 1 + 7.47T + 23T^{2} \)
29 \( 1 + 7.47iT - 29T^{2} \)
31 \( 1 - 9.09iT - 31T^{2} \)
37 \( 1 - 0.855iT - 37T^{2} \)
41 \( 1 - 2.19T + 41T^{2} \)
43 \( 1 + 0.954iT - 43T^{2} \)
47 \( 1 - 11.0iT - 47T^{2} \)
53 \( 1 - 3.09T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 + 8.05iT - 61T^{2} \)
67 \( 1 + 5.33iT - 67T^{2} \)
71 \( 1 + 6.43iT - 71T^{2} \)
73 \( 1 - 4.57T + 73T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 + 4.38iT - 83T^{2} \)
89 \( 1 + 4.28T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−8.635297363267731127980550424647, −7.81968246353340753848142999936, −7.49610477985976023466997541178, −6.31606778139457668427594965225, −5.83318339722536549544613670715, −4.72751963873125208702770308672, −4.33687265188735883967487149739, −3.43321487145730315726121216866, −2.06419079963782073435668343545, −1.57130652442184774428593308706, 0.76396263199844929371335935569, 2.01902356879183282387033617486, 3.00063871664315740460628072492, 3.89489850999459338074006353089, 4.65073855003156592545465248889, 5.59225842590097213902776386297, 5.99884395381678841385752248721, 7.00369301676377418664580681448, 7.78364447486668115392604563215, 8.548019188179154447439554001290

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