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Properties

Label 2-755-151.8-c1-0-38
Degree $2$
Conductor $755$
Sign $0.628 + 0.777i$
Analytic cond. $6.02870$
Root an. cond. $2.45534$
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  + 1.48·2-s + (0.0243 + 0.0750i)3-s + 0.198·4-s + (0.809 − 0.587i)5-s + (0.0361 + 0.111i)6-s + (−0.457 + 0.332i)7-s − 2.67·8-s + (2.42 − 1.75i)9-s + (1.19 − 0.871i)10-s + (1.35 − 4.16i)11-s + (0.00484 + 0.0149i)12-s + (−1.78 − 5.48i)13-s + (−0.677 + 0.492i)14-s + (0.0638 + 0.0463i)15-s − 4.35·16-s + (4.27 + 3.10i)17-s + ⋯
L(s)  = 1  + 1.04·2-s + (0.0140 + 0.0433i)3-s + 0.0993·4-s + (0.361 − 0.262i)5-s + (0.0147 + 0.0454i)6-s + (−0.172 + 0.125i)7-s − 0.944·8-s + (0.807 − 0.586i)9-s + (0.379 − 0.275i)10-s + (0.408 − 1.25i)11-s + (0.00139 + 0.00430i)12-s + (−0.494 − 1.52i)13-s + (−0.181 + 0.131i)14-s + (0.0164 + 0.0119i)15-s − 1.08·16-s + (1.03 + 0.753i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(755\)    =    \(5 \cdot 151\)
Sign: $0.628 + 0.777i$
Analytic conductor: \(6.02870\)
Root analytic conductor: \(2.45534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{755} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 755,\ (\ :1/2),\ 0.628 + 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23574 - 1.06769i\)
\(L(\frac12)\) \(\approx\) \(2.23574 - 1.06769i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 + 0.587i)T \)
151 \( 1 + (-12.1 + 1.88i)T \)
good2 \( 1 - 1.48T + 2T^{2} \)
3 \( 1 + (-0.0243 - 0.0750i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.457 - 0.332i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-1.35 + 4.16i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.78 + 5.48i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.27 - 3.10i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 - 0.970T + 19T^{2} \)
23 \( 1 - 6.80T + 23T^{2} \)
29 \( 1 + (2.46 - 7.59i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.316 - 0.230i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.50 + 7.71i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.10 - 6.48i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-1.17 - 0.851i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (0.973 + 2.99i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.71 + 8.35i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + 14.9T + 59T^{2} \)
61 \( 1 + (-3.17 + 9.76i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (2.94 + 2.13i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (10.3 - 7.51i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (6.90 - 5.01i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (8.05 - 5.84i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.49 - 3.26i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-8.53 - 6.19i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-7.04 - 5.11i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.24920399895081022664960008339, −9.333388382539094865373483225559, −8.665939805893005765754050061487, −7.51489156927530302254422932683, −6.33133904485072954052537442165, −5.61724033075068091289489684542, −4.92091994403881552764865819327, −3.56789539850650283798961801993, −3.11098734370567836826281766468, −1.00391753338751615947610845126, 1.80512033611355862709941080433, 3.03438405769576039123456259018, 4.39924758882150471494963865991, 4.72077124594043163410173883744, 5.90483501711406625580477693801, 7.00568377023653466273856716817, 7.39886014239140834031727200855, 9.089522921537542194078346189442, 9.590654137706340378984517077686, 10.36637113213065096785405037564

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