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Properties

Label 2-310-155.109-c1-0-9
Degree $2$
Conductor $310$
Sign $0.833 + 0.552i$
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  + (−0.587 − 0.809i)2-s + (−0.00820 + 0.0112i)3-s + (−0.309 + 0.951i)4-s + (1.91 − 1.15i)5-s + 0.0139·6-s + (1.51 + 0.493i)7-s + (0.951 − 0.309i)8-s + (0.926 + 2.85i)9-s + (−2.05 − 0.871i)10-s + (−0.445 + 1.37i)11-s + (−0.00820 − 0.0112i)12-s + (0.679 − 0.935i)13-s + (−0.493 − 1.51i)14-s + (−0.00269 + 0.0310i)15-s + (−0.809 − 0.587i)16-s + (−4.69 + 1.52i)17-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (−0.00473 + 0.00651i)3-s + (−0.154 + 0.475i)4-s + (0.856 − 0.515i)5-s + 0.00569·6-s + (0.574 + 0.186i)7-s + (0.336 − 0.109i)8-s + (0.308 + 0.950i)9-s + (−0.651 − 0.275i)10-s + (−0.134 + 0.413i)11-s + (−0.00236 − 0.00325i)12-s + (0.188 − 0.259i)13-s + (−0.131 − 0.406i)14-s + (−0.000694 + 0.00802i)15-s + (−0.202 − 0.146i)16-s + (−1.13 + 0.370i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310\)    =    \(2 \cdot 5 \cdot 31\)
Sign: $0.833 + 0.552i$
Analytic conductor: \(2.47536\)
Root analytic conductor: \(1.57332\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{310} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 310,\ (\ :1/2),\ 0.833 + 0.552i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23378 - 0.371420i\)
\(L(\frac12)\) \(\approx\) \(1.23378 - 0.371420i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 + (-1.91 + 1.15i)T \)
31 \( 1 + (0.639 + 5.53i)T \)
good3 \( 1 + (0.00820 - 0.0112i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-1.51 - 0.493i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.445 - 1.37i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.679 + 0.935i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.69 - 1.52i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-6.47 + 4.70i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-8.04 + 2.61i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (6.75 - 4.90i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 5.84iT - 37T^{2} \)
41 \( 1 + (-7.00 + 5.08i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (2.55 + 3.51i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (4.85 - 6.67i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (12.8 - 4.16i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.06 + 3.67i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 2.87T + 61T^{2} \)
67 \( 1 - 13.2iT - 67T^{2} \)
71 \( 1 + (1.59 + 4.90i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-13.2 - 4.30i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.87 - 8.85i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.82 + 6.63i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.41 - 7.42i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.07 + 0.999i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.21746421390793581114500390159, −10.87090289665290049568378717054, −9.617474357832014066593851793012, −9.015449761980291834936974714762, −7.969912188581804284867371391830, −6.91800547728844284338624441133, −5.29821254485812649092268172014, −4.63420722525493199926591429492, −2.68676314913035811488429830179, −1.51081030557556121725138252139, 1.48043309275926243888921984950, 3.34290712218279679275767237346, 4.98520714262158425002671749149, 6.03991532004406810331570071289, 6.91386284177859322160818042124, 7.82279885292181893686813356650, 9.251733248438329039500151880322, 9.521419021826439267906495206443, 10.85001415673841732894227915452, 11.44292815769463175617030360816

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