Properties

Label 2-310-5.4-c1-0-2
Degree $2$
Conductor $310$
Sign $-0.0410 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 1.55i·3-s − 4-s + (−2.23 + 0.0917i)5-s + 1.55·6-s + 4.87i·7-s i·8-s + 0.590·9-s + (−0.0917 − 2.23i)10-s + 3.14·11-s + 1.55i·12-s + 4.02i·13-s − 4.87·14-s + (0.142 + 3.46i)15-s + 16-s + 1.59i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.896i·3-s − 0.5·4-s + (−0.999 + 0.0410i)5-s + 0.633·6-s + 1.84i·7-s − 0.353i·8-s + 0.196·9-s + (−0.0290 − 0.706i)10-s + 0.947·11-s + 0.448i·12-s + 1.11i·13-s − 1.30·14-s + (0.0367 + 0.895i)15-s + 0.250·16-s + 0.385i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0410 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0410 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310\)    =    \(2 \cdot 5 \cdot 31\)
Sign: $-0.0410 - 0.999i$
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: \((2,\ 310,\ (\ :1/2),\ -0.0410 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.697160 + 0.726378i\)
\(L(\frac12)\) \(\approx\) \(0.697160 + 0.726378i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.0917i)T \)
31 \( 1 - T \)
good3 \( 1 + 1.55iT - 3T^{2} \)
7 \( 1 - 4.87iT - 7T^{2} \)
11 \( 1 - 3.14T + 11T^{2} \)
13 \( 1 - 4.02iT - 13T^{2} \)
17 \( 1 - 1.59iT - 17T^{2} \)
19 \( 1 + 3.28T + 19T^{2} \)
23 \( 1 - 6.16iT - 23T^{2} \)
29 \( 1 - 2.14T + 29T^{2} \)
37 \( 1 - 5.55iT - 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + 1.81iT - 43T^{2} \)
47 \( 1 + 6.26iT - 47T^{2} \)
53 \( 1 + 12.9iT - 53T^{2} \)
59 \( 1 - 14.6T + 59T^{2} \)
61 \( 1 + 6.79T + 61T^{2} \)
67 \( 1 + 2.67iT - 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 - 4.15iT - 73T^{2} \)
79 \( 1 - 5.10T + 79T^{2} \)
83 \( 1 - 2.73iT - 83T^{2} \)
89 \( 1 - 2.00T + 89T^{2} \)
97 \( 1 - 15.9iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.91247404228361450208007173167, −11.57736856673407210871373961278, −9.719334667534625113420790548722, −8.692029044392346569647102946037, −8.201552635349377764073352896868, −6.88321874389846221800173433471, −6.42364892751756232129331203964, −5.06714257986021403749122812034, −3.73905994944898685905500697611, −1.88514707584235547361292973645, 0.797071585431267279857302228674, 3.35622130248904037507948230898, 4.14063734308090964011070777509, 4.71883319792441897598133474416, 6.73542853997715437802308434140, 7.71481519578964045066955021329, 8.764429708811378138660486849509, 9.970615843410624324525767555920, 10.58039703500463210948392519420, 11.10436067541549838381936068296

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