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Properties

Label 2-310-5.4-c1-0-6
Degree $2$
Conductor $310$
Sign $0.980 + 0.195i$
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 − 0.287i·3-s − 4-s + (−0.437 + 2.19i)5-s − 0.287·6-s + 1.04i·7-s + i·8-s + 2.91·9-s + (2.19 + 0.437i)10-s + 3.63·11-s + 0.287i·12-s + 1.41i·13-s + 1.04·14-s + (0.630 + 0.125i)15-s + 16-s − 3.91i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.165i·3-s − 0.5·4-s + (−0.195 + 0.980i)5-s − 0.117·6-s + 0.394i·7-s + 0.353i·8-s + 0.972·9-s + (0.693 + 0.138i)10-s + 1.09·11-s + 0.0829i·12-s + 0.391i·13-s + 0.278·14-s + (0.162 + 0.0324i)15-s + 0.250·16-s − 0.950i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310\)    =    \(2 \cdot 5 \cdot 31\)
Sign: $0.980 + 0.195i$
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: \((2,\ 310,\ (\ :1/2),\ 0.980 + 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31635 - 0.129939i\)
\(L(\frac12)\) \(\approx\) \(1.31635 - 0.129939i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.437 - 2.19i)T \)
31 \( 1 - T \)
good3 \( 1 + 0.287iT - 3T^{2} \)
7 \( 1 - 1.04iT - 7T^{2} \)
11 \( 1 - 3.63T + 11T^{2} \)
13 \( 1 - 1.41iT - 13T^{2} \)
17 \( 1 + 3.91iT - 17T^{2} \)
19 \( 1 - 4.96T + 19T^{2} \)
23 \( 1 - 8.00iT - 23T^{2} \)
29 \( 1 + 6.59T + 29T^{2} \)
37 \( 1 + 3.71iT - 37T^{2} \)
41 \( 1 - 2.74T + 41T^{2} \)
43 \( 1 - 3.73iT - 43T^{2} \)
47 \( 1 + 2.35iT - 47T^{2} \)
53 \( 1 - 0.335iT - 53T^{2} \)
59 \( 1 + 4.33T + 59T^{2} \)
61 \( 1 + 8.33T + 61T^{2} \)
67 \( 1 + 8.27iT - 67T^{2} \)
71 \( 1 + 5.59T + 71T^{2} \)
73 \( 1 + 8.43iT - 73T^{2} \)
79 \( 1 - 1.42T + 79T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + 16.4iT - 97T^{2} \)
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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

−11.64472432760279515291649857378, −10.94880652361043597141386902734, −9.609576274029393517041302991530, −9.365351989279441715349952440821, −7.63691473210040009973289761050, −6.98884610064339837903991823269, −5.67268527056690203227759020433, −4.17779458516582549589139341537, −3.16951795692188074473803437244, −1.65444680949152267990034502378, 1.21209082254864305147923445681, 3.84655997027808174360021860771, 4.56023725220874372782715110565, 5.78347897089306501919210248860, 6.93063592659653336043272082075, 7.86428643961515445002350242723, 8.843643615556759329297684210013, 9.637682883163733632887451167865, 10.60091069159777763543910825235, 11.95804427361975966894425130204

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