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Properties

Label 2-750-25.11-c1-0-13
Degree $2$
Conductor $750$
Sign $-0.187 + 0.982i$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)6-s + 1.72·7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (1.97 − 1.43i)11-s + (−0.809 − 0.587i)12-s + (−2.40 − 1.74i)13-s + (1.39 − 1.01i)14-s + (−0.809 − 0.587i)16-s + (−1.03 − 3.18i)17-s − 0.999·18-s + (0.694 + 2.13i)19-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (−0.126 − 0.388i)6-s + 0.652·7-s + (−0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + (0.595 − 0.432i)11-s + (−0.233 − 0.169i)12-s + (−0.666 − 0.484i)13-s + (0.373 − 0.271i)14-s + (−0.202 − 0.146i)16-s + (−0.250 − 0.771i)17-s − 0.235·18-s + (0.159 + 0.490i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.187 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46554 - 1.77153i\)
\(L(\frac12)\) \(\approx\) \(1.46554 - 1.77153i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 \)
good7 \( 1 - 1.72T + 7T^{2} \)
11 \( 1 + (-1.97 + 1.43i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.40 + 1.74i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.03 + 3.18i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.694 - 2.13i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-7.37 + 5.35i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.89 - 8.91i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.89 + 5.82i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.67 - 1.94i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.95 - 3.60i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.06T + 43T^{2} \)
47 \( 1 + (-1.54 + 4.74i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.11 + 6.51i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.147 - 0.107i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.16 - 3.02i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.67 - 11.3i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.51 - 10.8i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.44 + 3.95i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.50 - 10.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.82 - 8.69i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-6.56 + 4.77i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.05 + 9.40i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.26499253892338593818216072331, −9.239217839085251635111737943729, −8.436227246887676208731192774841, −7.37649734424147841180067063931, −6.63907253246798419157100698411, −5.47097884642711357925774555429, −4.70651104862278520056304430005, −3.41056128865499857893186839027, −2.39432410281391021779316309777, −1.04075997064099875422890647544, 1.91543962907373329555912867210, 3.31025095032694638901118378278, 4.39391060413323703954393623504, 4.99302696164616838846419719866, 6.09116777925122335226808004826, 7.14902774454321612704066274254, 7.85721371860222511979912960455, 9.000168447081126891459849708932, 9.511346983118703950485625370638, 10.75703850510693853819014125203

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