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Properties

Label 2-750-125.104-c1-0-21
Degree $2$
Conductor $750$
Sign $0.0671 + 0.997i$
Analytic cond. $5.98878$
Root an. cond. $2.44719$
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  + (−0.248 − 0.968i)2-s + (0.982 − 0.187i)3-s + (−0.876 + 0.481i)4-s + (1.73 + 1.41i)5-s + (−0.425 − 0.904i)6-s + (2.54 − 3.50i)7-s + (0.684 + 0.728i)8-s + (0.929 − 0.368i)9-s + (0.940 − 2.02i)10-s + (−5.83 + 1.49i)11-s + (−0.770 + 0.637i)12-s + (−1.91 − 4.83i)13-s + (−4.03 − 1.59i)14-s + (1.96 + 1.06i)15-s + (0.535 − 0.844i)16-s + (1.66 − 3.02i)17-s + ⋯
L(s)  = 1  + (−0.175 − 0.684i)2-s + (0.567 − 0.108i)3-s + (−0.438 + 0.240i)4-s + (0.774 + 0.633i)5-s + (−0.173 − 0.369i)6-s + (0.963 − 1.32i)7-s + (0.242 + 0.257i)8-s + (0.309 − 0.122i)9-s + (0.297 − 0.641i)10-s + (−1.75 + 0.451i)11-s + (−0.222 + 0.184i)12-s + (−0.531 − 1.34i)13-s + (−1.07 − 0.426i)14-s + (0.507 + 0.275i)15-s + (0.133 − 0.211i)16-s + (0.403 − 0.734i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0671 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0671 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.0671 + 0.997i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.0671 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36304 - 1.27437i\)
\(L(\frac12)\) \(\approx\) \(1.36304 - 1.27437i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.248 + 0.968i)T \)
3 \( 1 + (-0.982 + 0.187i)T \)
5 \( 1 + (-1.73 - 1.41i)T \)
good7 \( 1 + (-2.54 + 3.50i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (5.83 - 1.49i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (1.91 + 4.83i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (-1.66 + 3.02i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (-0.571 + 2.99i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (-9.04 - 0.568i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (-1.72 - 0.218i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (0.253 + 0.139i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (-3.11 - 1.97i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (-0.514 - 8.18i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (5.88 - 1.91i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-7.85 + 8.36i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (3.00 + 1.41i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (-2.99 - 3.61i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (0.504 - 8.02i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (-0.471 - 3.73i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (2.81 + 2.63i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (4.12 + 3.41i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (0.974 + 5.10i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (12.1 + 2.31i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (11.4 - 13.8i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (1.81 - 14.3i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.32608725789961843989164480763, −9.623052248047204870287771137400, −8.380009713152739302585712833553, −7.49973882877419620114898648410, −7.17478209150119906791609531412, −5.28542747478869464569092163718, −4.75708405785166374836024816648, −3.09538883735157616128294368756, −2.56826616728265359682763045812, −1.03122947671578571363819706904, 1.73049256416801123991433070754, 2.72092493696107358562792532848, 4.58116147654053502336691044462, 5.28593103002069110113597421769, 5.90371886494821752818386852990, 7.27473743891999680888538521301, 8.243373159614485133528087220492, 8.700308850251432523322943217177, 9.391629506121292993051362339985, 10.31222205782368346132007323972

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