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Properties

Label 2-310-155.108-c1-0-7
Degree $2$
Conductor $310$
Sign $0.900 + 0.434i$
Analytic cond. $2.47536$
Root an. cond. $1.57332$
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  + (−0.156 + 0.987i)2-s + (−2.26 + 0.358i)3-s + (−0.951 − 0.309i)4-s + (−1.29 + 1.82i)5-s − 2.29i·6-s + (−2.04 − 4.01i)7-s + (0.453 − 0.891i)8-s + (2.14 − 0.696i)9-s + (−1.60 − 1.56i)10-s + (3.76 + 1.22i)11-s + (2.26 + 0.358i)12-s + (4.43 − 0.702i)13-s + (4.28 − 1.39i)14-s + (2.26 − 4.59i)15-s + (0.809 + 0.587i)16-s + (−0.689 + 1.35i)17-s + ⋯
L(s)  = 1  + (−0.110 + 0.698i)2-s + (−1.30 + 0.207i)3-s + (−0.475 − 0.154i)4-s + (−0.577 + 0.816i)5-s − 0.935i·6-s + (−0.773 − 1.51i)7-s + (0.160 − 0.315i)8-s + (0.714 − 0.232i)9-s + (−0.506 − 0.493i)10-s + (1.13 + 0.368i)11-s + (0.653 + 0.103i)12-s + (1.23 − 0.194i)13-s + (1.14 − 0.372i)14-s + (0.585 − 1.18i)15-s + (0.202 + 0.146i)16-s + (−0.167 + 0.328i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310\)    =    \(2 \cdot 5 \cdot 31\)
Sign: $0.900 + 0.434i$
Analytic conductor: \(2.47536\)
Root analytic conductor: \(1.57332\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{310} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 310,\ (\ :1/2),\ 0.900 + 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.478853 - 0.109511i\)
\(L(\frac12)\) \(\approx\) \(0.478853 - 0.109511i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.156 - 0.987i)T \)
5 \( 1 + (1.29 - 1.82i)T \)
31 \( 1 + (-5.42 + 1.24i)T \)
good3 \( 1 + (2.26 - 0.358i)T + (2.85 - 0.927i)T^{2} \)
7 \( 1 + (2.04 + 4.01i)T + (-4.11 + 5.66i)T^{2} \)
11 \( 1 + (-3.76 - 1.22i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (-4.43 + 0.702i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (0.689 - 1.35i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (2.65 + 3.65i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (3.67 + 1.87i)T + (13.5 + 18.6i)T^{2} \)
29 \( 1 + (-3.51 + 2.55i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + (5.91 + 5.91i)T + 37iT^{2} \)
41 \( 1 + (-3.28 + 2.38i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-1.60 + 10.1i)T + (-40.8 - 13.2i)T^{2} \)
47 \( 1 + (3.57 - 0.566i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (3.49 + 1.77i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-2.45 + 3.37i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 - 9.90iT - 61T^{2} \)
67 \( 1 + (2.92 - 2.92i)T - 67iT^{2} \)
71 \( 1 + (3.95 + 12.1i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.390 - 0.766i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (2.07 + 6.39i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.632 - 3.99i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (1.22 - 3.77i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.53 - 12.8i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.41946808661407453316986394591, −10.59442556023776861945516796210, −10.15122463629317924941085266025, −8.693342759762445985518919154111, −7.36113407937478471207844365063, −6.53539473040774829894792980379, −6.18940770799062138527069436164, −4.40071640633358929447827388490, −3.77127691996015909068608061697, −0.50179313048605343194129933759, 1.32463473690447076163722695806, 3.38900686358557508324528375285, 4.68703047700624330373333250697, 5.91782652823150362524120137129, 6.40663496415337523205046184309, 8.372838765139070088081388829373, 8.914785447641067787609447540407, 9.964444874225095342092907478522, 11.33616519696670721848489408020, 11.70353175703749381162647530055

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