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Properties

Label 2-750-25.6-c1-0-3
Degree $2$
Conductor $750$
Sign $-0.484 - 0.874i$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s + 2.61·7-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.0883 − 0.271i)11-s + (0.309 − 0.951i)12-s + (−0.866 + 2.66i)13-s + (0.809 + 2.49i)14-s + (0.309 − 0.951i)16-s + (5.25 + 3.81i)17-s + 18-s + (0.358 + 0.260i)19-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.330 − 0.239i)6-s + 0.990·7-s + (−0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + (−0.0266 − 0.0819i)11-s + (0.0892 − 0.274i)12-s + (−0.240 + 0.739i)13-s + (0.216 + 0.665i)14-s + (0.0772 − 0.237i)16-s + (1.27 + 0.925i)17-s + 0.235·18-s + (0.0822 + 0.0597i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.484 - 0.874i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.484 - 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738248 + 1.25256i\)
\(L(\frac12)\) \(\approx\) \(0.738248 + 1.25256i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 \)
good7 \( 1 - 2.61T + 7T^{2} \)
11 \( 1 + (0.0883 + 0.271i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.866 - 2.66i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-5.25 - 3.81i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.358 - 0.260i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.01 - 6.20i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (5.87 - 4.26i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.93 + 2.13i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.40 + 4.31i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.64 + 5.06i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.05T + 43T^{2} \)
47 \( 1 + (7.27 - 5.28i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (11.2 - 8.17i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.15 - 3.56i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.02 - 12.3i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-8.27 - 6.01i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-3.89 + 2.83i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.64 + 11.2i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.32 + 4.59i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.17 - 3.03i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.90 + 5.86i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.40 - 3.20i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.77763465958434060632724123122, −9.622401035702901449175874336643, −8.954112501250037353479148427804, −7.78886014683973540769962974250, −7.31711071958885957194862034013, −5.97247030686195297817052922494, −5.40783891462949503559364660512, −4.43280131316825109465397934777, −3.49189548187274873075726266961, −1.56800621892576639991592459765, 0.807822671293984596481150103567, 2.15527826531034376542665911857, 3.38759143232703157999786883478, 4.82842532717074968917859337406, 5.25986709140660056399081218745, 6.43836529957462184488610120566, 7.65574062956991679274476052147, 8.202583983297475251112321711138, 9.486453712168909092543872312535, 10.19291703640824825339230474160

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