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Properties

Label 2-755-151.19-c1-0-31
Degree $2$
Conductor $755$
Sign $0.714 + 0.699i$
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  − 1.95·2-s + (−0.0899 + 0.276i)3-s + 1.82·4-s + (0.809 + 0.587i)5-s + (0.175 − 0.541i)6-s + (0.856 + 0.622i)7-s + 0.346·8-s + (2.35 + 1.71i)9-s + (−1.58 − 1.14i)10-s + (−1.88 − 5.81i)11-s + (−0.163 + 0.504i)12-s + (1.46 − 4.49i)13-s + (−1.67 − 1.21i)14-s + (−0.235 + 0.171i)15-s − 4.32·16-s + (0.588 − 0.427i)17-s + ⋯
L(s)  = 1  − 1.38·2-s + (−0.0519 + 0.159i)3-s + 0.911·4-s + (0.361 + 0.262i)5-s + (0.0718 − 0.221i)6-s + (0.323 + 0.235i)7-s + 0.122·8-s + (0.786 + 0.571i)9-s + (−0.500 − 0.363i)10-s + (−0.569 − 1.75i)11-s + (−0.0473 + 0.145i)12-s + (0.405 − 1.24i)13-s + (−0.447 − 0.325i)14-s + (−0.0608 + 0.0441i)15-s − 1.08·16-s + (0.142 − 0.103i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(755\)    =    \(5 \cdot 151\)
Sign: $0.714 + 0.699i$
Analytic conductor: \(6.02870\)
Root analytic conductor: \(2.45534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{755} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 755,\ (\ :1/2),\ 0.714 + 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.710975 - 0.290116i\)
\(L(\frac12)\) \(\approx\) \(0.710975 - 0.290116i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.03 - 11.9i)T \)
good2 \( 1 + 1.95T + 2T^{2} \)
3 \( 1 + (0.0899 - 0.276i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-0.856 - 0.622i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (1.88 + 5.81i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.46 + 4.49i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.588 + 0.427i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + 3.86T + 19T^{2} \)
23 \( 1 - 6.49T + 23T^{2} \)
29 \( 1 + (0.889 + 2.73i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.86 - 4.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.63 + 11.1i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.63 + 5.02i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (4.60 - 3.34i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (0.719 - 2.21i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.785 - 2.41i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 - 9.34T + 59T^{2} \)
61 \( 1 + (1.40 + 4.32i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-8.26 + 6.00i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-0.898 - 0.652i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (6.61 + 4.80i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-8.17 - 5.93i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.96 + 4.33i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.09 + 1.52i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-13.0 + 9.46i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.35499082742643355423543871221, −9.293992703460049626165220617537, −8.557689605496040985145295579505, −7.931095543082434931998623155936, −7.09969016345361073014518166487, −5.86789410125962582733958096233, −5.04247916638336176134972409851, −3.45955614163273539702910838455, −2.14777635104343138351238053382, −0.70429312494237417132021081566, 1.32045602319735002230287435057, 2.10103128945256377805388455296, 4.18602289754866238289089828840, 4.90970159555547471548045741704, 6.62843802337515342930940684428, 7.07148799152172667397119124596, 7.950366953817577166442254479470, 8.945201505715692396815002463565, 9.568730866398125747572949945431, 10.13159266907151658127399156614

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