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Properties

Label 2-310-155.99-c2-0-9
Degree $2$
Conductor $310$
Sign $0.153 - 0.988i$
Analytic cond. $8.44688$
Root an. cond. $2.90635$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + (2.70 + 4.69i)3-s − 2.00·4-s + (2.12 − 4.52i)5-s + (6.63 − 3.82i)6-s + (−7.85 + 4.53i)7-s + 2.82i·8-s + (−10.1 + 17.6i)9-s + (−6.39 − 3.00i)10-s + (11.8 + 6.86i)11-s + (−5.41 − 9.38i)12-s + (−10.6 + 18.4i)13-s + (6.41 + 11.1i)14-s + (26.9 − 2.27i)15-s + 4.00·16-s + (2.30 + 3.98i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.902 + 1.56i)3-s − 0.500·4-s + (0.425 − 0.905i)5-s + (1.10 − 0.638i)6-s + (−1.12 + 0.647i)7-s + 0.353i·8-s + (−1.12 + 1.95i)9-s + (−0.639 − 0.300i)10-s + (1.08 + 0.623i)11-s + (−0.451 − 0.781i)12-s + (−0.820 + 1.42i)13-s + (0.458 + 0.793i)14-s + (1.79 − 0.151i)15-s + 0.250·16-s + (0.135 + 0.234i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310\)    =    \(2 \cdot 5 \cdot 31\)
Sign: $0.153 - 0.988i$
Analytic conductor: \(8.44688\)
Root analytic conductor: \(2.90635\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{310} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 310,\ (\ :1),\ 0.153 - 0.988i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.35882 + 1.16356i\)
\(L(\frac12)\) \(\approx\) \(1.35882 + 1.16356i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
5 \( 1 + (-2.12 + 4.52i)T \)
31 \( 1 + (-1.23 - 30.9i)T \)
good3 \( 1 + (-2.70 - 4.69i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (7.85 - 4.53i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-11.8 - 6.86i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (10.6 - 18.4i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-2.30 - 3.98i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (7.69 + 13.3i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 - 34.1T + 529T^{2} \)
29 \( 1 - 19.5iT - 841T^{2} \)
37 \( 1 + (-24.4 - 42.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-18.9 + 32.8i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (40.0 + 69.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 0.952iT - 2.20e3T^{2} \)
53 \( 1 + (-11.2 + 19.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-2.38 - 4.13i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + 5.24iT - 3.72e3T^{2} \)
67 \( 1 + (-29.1 - 16.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-25.1 + 43.4i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-18.8 + 32.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (20.7 - 12.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-46.9 + 81.3i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 45.2iT - 7.92e3T^{2} \)
97 \( 1 - 139. iT - 9.40e3T^{2} \)
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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

−11.68608712357617833376482845315, −10.38567485001599519091881105576, −9.588787804488798073128261187611, −9.070620631698684974694242207070, −8.787622804212436864810112865550, −6.75402954211201552488046203849, −5.09721297518346282722898553983, −4.42309666992098206215887963564, −3.32289010895118417991120785530, −2.10639055593008329116901325890, 0.78238131657483121860358764878, 2.71746177195163340370977029439, 3.55137922431685655465886999329, 5.95850041142991105740657857817, 6.53104434788820449382842601994, 7.33533922688603716385886048056, 7.993778138361915766477658075065, 9.251187495888821517549412775653, 9.975051806063840174517408232020, 11.38944698856786153010959189582

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