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Properties

Label 2-630-7.4-c3-0-23
Degree $2$
Conductor $630$
Sign $0.701 - 0.712i$
Analytic cond. $37.1712$
Root an. cond. $6.09681$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (−2.5 − 4.33i)5-s + (17.5 − 6.06i)7-s − 7.99·8-s + (5 − 8.66i)10-s + (−15 + 25.9i)11-s + 44·13-s + (28 + 24.2i)14-s + (−8 − 13.8i)16-s + (−12 + 20.7i)17-s + (−1 − 1.73i)19-s + 20·20-s − 60·22-s + (−91.5 − 158. i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + (0.944 − 0.327i)7-s − 0.353·8-s + (0.158 − 0.273i)10-s + (−0.411 + 0.712i)11-s + 0.938·13-s + (0.534 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (−0.171 + 0.296i)17-s + (−0.0120 − 0.0209i)19-s + 0.223·20-s − 0.581·22-s + (−0.829 − 1.43i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(37.1712\)
Root analytic conductor: \(6.09681\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :3/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.543971769\)
\(L(\frac12)\) \(\approx\) \(2.543971769\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 \)
5 \( 1 + (2.5 + 4.33i)T \)
7 \( 1 + (-17.5 + 6.06i)T \)
good11 \( 1 + (15 - 25.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 44T + 2.19e3T^{2} \)
17 \( 1 + (12 - 20.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (91.5 + 158. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 279T + 2.43e4T^{2} \)
31 \( 1 + (-20 + 34.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-38 - 65.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 423T + 6.89e4T^{2} \)
43 \( 1 - 305T + 7.95e4T^{2} \)
47 \( 1 + (-228 - 394. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (99 - 171. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (231 - 400. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (140.5 + 243. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-249.5 + 432. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 534T + 3.57e5T^{2} \)
73 \( 1 + (400 - 692. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-395 - 684. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 597T + 5.71e5T^{2} \)
89 \( 1 + (-508.5 - 880. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.33e3T + 9.12e5T^{2} \)
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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

−10.44036956479093150090084628201, −9.209950316584651949676264299241, −8.232607474680359693327436625711, −7.84821625994665184862736374012, −6.68314713074017526391227226304, −5.78136057399673434148131230344, −4.56584337808747722924790629743, −4.20258521716643431033506891646, −2.50680213697799396656639707472, −0.959338957016695649368711084979, 0.901833151838048543916732854087, 2.22460355670819254920368443430, 3.33960264766123525148228134886, 4.35764155222816492773749176997, 5.45766189140610791953553699123, 6.22504316544959545785195074158, 7.59860192235380961379470156168, 8.374423719746909737265586710385, 9.238056668476711312902644630016, 10.41290157805451205013015632470

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