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Properties

Label 2-310-155.129-c1-0-8
Degree $2$
Conductor $310$
Sign $0.579 + 0.815i$
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 + (−1.50 − 0.866i)3-s − 4-s + (2.21 + 0.294i)5-s + (0.866 − 1.50i)6-s + (−2.92 − 1.68i)7-s − i·8-s + (0.00257 + 0.00446i)9-s + (−0.294 + 2.21i)10-s + (0.186 + 0.323i)11-s + (1.50 + 0.866i)12-s + (4.86 − 2.80i)13-s + (1.68 − 2.92i)14-s + (−3.07 − 2.36i)15-s + 16-s + (−6.58 − 3.79i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.866 − 0.500i)3-s − 0.5·4-s + (0.991 + 0.131i)5-s + (0.353 − 0.612i)6-s + (−1.10 − 0.637i)7-s − 0.353i·8-s + (0.000858 + 0.00148i)9-s + (−0.0930 + 0.700i)10-s + (0.0562 + 0.0974i)11-s + (0.433 + 0.250i)12-s + (1.34 − 0.778i)13-s + (0.450 − 0.781i)14-s + (−0.793 − 0.610i)15-s + 0.250·16-s + (−1.59 − 0.921i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.579 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.579 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310\)    =    \(2 \cdot 5 \cdot 31\)
Sign: $0.579 + 0.815i$
Analytic conductor: \(2.47536\)
Root analytic conductor: \(1.57332\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{310} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 310,\ (\ :1/2),\ 0.579 + 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753230 - 0.388669i\)
\(L(\frac12)\) \(\approx\) \(0.753230 - 0.388669i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 - 0.294i)T \)
31 \( 1 + (3.26 + 4.50i)T \)
good3 \( 1 + (1.50 + 0.866i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.92 + 1.68i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.186 - 0.323i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.86 + 2.80i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (6.58 + 3.79i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.17 + 5.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.79iT - 23T^{2} \)
29 \( 1 - 4.59T + 29T^{2} \)
37 \( 1 + (-8.27 - 4.77i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.70 + 2.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.04 + 4.64i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.79iT - 47T^{2} \)
53 \( 1 + (3.48 - 2.01i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.56 - 4.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 12.9T + 61T^{2} \)
67 \( 1 + (-3.13 + 1.81i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.14 - 7.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.20 - 3.58i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.0564 - 0.0978i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.93 + 4.00i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.06T + 89T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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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

−11.44730107811303274180118834203, −10.64344415572890145054739385993, −9.557787551705120774104962787142, −8.829622385490628492083266886817, −7.21501678846700675997266275614, −6.48289370151199886238139093932, −6.02392127459969318798145841199, −4.78008141742603379118148537928, −3.06611651437628719078624258427, −0.70132970242432185023916672400, 1.84606643587833743949161727139, 3.43517554464096079986153797272, 4.76016517893244813942254870451, 6.04563739482787546590531550480, 6.29198039569359355520151987049, 8.506311323518113643091169450546, 9.279078474512528554174809620584, 10.10296811875753187703324797625, 10.87412912969583310739815755337, 11.65702566867119925662451654745

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