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Properties

Label 2-310-5.4-c1-0-5
Degree $2$
Conductor $310$
Sign $0.500 - 0.865i$
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  − i·2-s + 2.51i·3-s − 4-s + (1.93 + 1.11i)5-s + 2.51·6-s − 0.461i·7-s + i·8-s − 3.33·9-s + (1.11 − 1.93i)10-s + 0.183·11-s − 2.51i·12-s + 3.35i·13-s − 0.461·14-s + (−2.81 + 4.87i)15-s + 16-s + 2.33i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.45i·3-s − 0.5·4-s + (0.865 + 0.500i)5-s + 1.02·6-s − 0.174i·7-s + 0.353i·8-s − 1.11·9-s + (0.353 − 0.612i)10-s + 0.0552·11-s − 0.726i·12-s + 0.930i·13-s − 0.123·14-s + (−0.727 + 1.25i)15-s + 0.250·16-s + 0.565i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310\)    =    \(2 \cdot 5 \cdot 31\)
Sign: $0.500 - 0.865i$
Analytic conductor: \(2.47536\)
Root analytic conductor: \(1.57332\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{310} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 310,\ (\ :1/2),\ 0.500 - 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18063 + 0.681158i\)
\(L(\frac12)\) \(\approx\) \(1.18063 + 0.681158i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (-1.93 - 1.11i)T \)
31 \( 1 - T \)
good3 \( 1 - 2.51iT - 3T^{2} \)
7 \( 1 + 0.461iT - 7T^{2} \)
11 \( 1 - 0.183T + 11T^{2} \)
13 \( 1 - 3.35iT - 13T^{2} \)
17 \( 1 - 2.33iT - 17T^{2} \)
19 \( 1 + 2.79T + 19T^{2} \)
23 \( 1 + 1.25iT - 23T^{2} \)
29 \( 1 - 4.61T + 29T^{2} \)
37 \( 1 + 6.51iT - 37T^{2} \)
41 \( 1 - 0.376T + 41T^{2} \)
43 \( 1 + 9.42iT - 43T^{2} \)
47 \( 1 + 2.13iT - 47T^{2} \)
53 \( 1 + 11.0iT - 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 + 1.80iT - 67T^{2} \)
71 \( 1 + 5.18T + 71T^{2} \)
73 \( 1 - 11.5iT - 73T^{2} \)
79 \( 1 - 7.03T + 79T^{2} \)
83 \( 1 + 6.70iT - 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + 12.3iT - 97T^{2} \)
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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

−11.49565674077933667391911699772, −10.61526990240047147569843390714, −10.19982491208767713616217735800, −9.325501979151479261598798654360, −8.604480820729680547560903607295, −6.82215127473188124029950388796, −5.60916499830207894523097360803, −4.48922672463026160604374198871, −3.59040356676620555176075355836, −2.17376073749496433723391921040, 1.10286945066020886871114462058, 2.65236140839397967119995475184, 4.80892645452724957872668366752, 5.92679340773144745303624238240, 6.55664550308075503902705399816, 7.66727098416026279366713047475, 8.399820118993767912688682565541, 9.390658872292197222563811675427, 10.49401162381309525604544202787, 11.96846438118854896569971155155

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