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Properties

Label 2-310-31.16-c1-0-7
Degree $2$
Conductor $310$
Sign $-0.696 + 0.717i$
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  + (0.809 − 0.587i)2-s + (−1.04 − 0.757i)3-s + (0.309 − 0.951i)4-s − 5-s − 1.28·6-s + (0.253 − 0.781i)7-s + (−0.309 − 0.951i)8-s + (−0.414 − 1.27i)9-s + (−0.809 + 0.587i)10-s + (0.565 − 1.73i)11-s + (−1.04 + 0.757i)12-s + (−3.69 − 2.68i)13-s + (−0.253 − 0.781i)14-s + (1.04 + 0.757i)15-s + (−0.809 − 0.587i)16-s + (0.168 + 0.517i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.601 − 0.437i)3-s + (0.154 − 0.475i)4-s − 0.447·5-s − 0.525·6-s + (0.0959 − 0.295i)7-s + (−0.109 − 0.336i)8-s + (−0.138 − 0.424i)9-s + (−0.255 + 0.185i)10-s + (0.170 − 0.524i)11-s + (−0.300 + 0.218i)12-s + (−1.02 − 0.744i)13-s + (−0.0678 − 0.208i)14-s + (0.269 + 0.195i)15-s + (−0.202 − 0.146i)16-s + (0.0407 + 0.125i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 + 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310\)    =    \(2 \cdot 5 \cdot 31\)
Sign: $-0.696 + 0.717i$
Analytic conductor: \(2.47536\)
Root analytic conductor: \(1.57332\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{310} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 310,\ (\ :1/2),\ -0.696 + 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446595 - 1.05500i\)
\(L(\frac12)\) \(\approx\) \(0.446595 - 1.05500i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + T \)
31 \( 1 + (-2.53 - 4.95i)T \)
good3 \( 1 + (1.04 + 0.757i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-0.253 + 0.781i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.565 + 1.73i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (3.69 + 2.68i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.168 - 0.517i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.203 - 0.148i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.571 + 1.75i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.67 + 3.39i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 4.66T + 37T^{2} \)
41 \( 1 + (-8.31 + 6.03i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (2.28 - 1.66i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-2.95 - 2.14i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.25 + 13.0i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.88 - 5.72i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 0.485T + 61T^{2} \)
67 \( 1 - 2.59T + 67T^{2} \)
71 \( 1 + (1.25 + 3.85i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.187 - 0.578i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.42 - 7.45i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (11.3 - 8.24i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.198 - 0.609i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (2.38 - 7.33i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.51745778556043333265311980791, −10.70026545840124243910307967955, −9.728998206926417858646638475329, −8.406570414990565443880627165713, −7.26875226145211091365424317806, −6.30014905918919733662069252903, −5.30413636060156024969745951973, −4.09979857484342198078568426232, −2.80415537194797071713977598451, −0.75388071693572200319522501473, 2.51381095372997217117526101194, 4.25010021694154963347648639659, 4.89366590768701552786660838050, 5.98614147337050546473119826597, 7.13438415311292897939152910471, 7.999817428691417346766031807198, 9.256895573545327898903804920975, 10.26952783572592672485049782965, 11.41587821430068997823678952160, 11.90840462148159375421152586075

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