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Properties

Label 2-312-13.4-c1-0-3
Degree $2$
Conductor $312$
Sign $0.252 + 0.967i$
Analytic cond. $2.49133$
Root an. cond. $1.57839$
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  + (0.5 − 0.866i)3-s − 1.73i·5-s + (0.633 − 0.366i)7-s + (−0.499 − 0.866i)9-s + (−0.633 − 0.366i)11-s + (2.59 − 2.5i)13-s + (−1.49 − 0.866i)15-s + (−1.86 − 3.23i)17-s + (−1.09 + 0.633i)19-s − 0.732i·21-s + (−1.09 + 1.90i)23-s + 2.00·25-s − 0.999·27-s + (2.96 − 5.13i)29-s + 5.46i·31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s − 0.774i·5-s + (0.239 − 0.138i)7-s + (−0.166 − 0.288i)9-s + (−0.191 − 0.110i)11-s + (0.720 − 0.693i)13-s + (−0.387 − 0.223i)15-s + (−0.452 − 0.783i)17-s + (−0.251 + 0.145i)19-s − 0.159i·21-s + (−0.228 + 0.396i)23-s + 0.400·25-s − 0.192·27-s + (0.550 − 0.953i)29-s + 0.981i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ 0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11986 - 0.865021i\)
\(L(\frac12)\) \(\approx\) \(1.11986 - 0.865021i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-2.59 + 2.5i)T \)
good5 \( 1 + 1.73iT - 5T^{2} \)
7 \( 1 + (-0.633 + 0.366i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.633 + 0.366i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.86 + 3.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.09 - 0.633i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.09 - 1.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.96 + 5.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.46iT - 31T^{2} \)
37 \( 1 + (-7.33 - 4.23i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.33 + 2.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.901 - 1.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.73iT - 47T^{2} \)
53 \( 1 - 5.92T + 53T^{2} \)
59 \( 1 + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.86 - 8.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.56 + 2.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.09 - 4.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.19iT - 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 - 15.6iT - 83T^{2} \)
89 \( 1 + (2.19 + 1.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.66 - 5i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.59557652847469844457841940545, −10.60810591121783212016986954346, −9.447037610849405893838903346236, −8.504665455673927040210727836566, −7.85670532852442204936272980019, −6.63709505972233957001980735109, −5.49479501976618177001612046759, −4.34488193542714438111454445035, −2.82790479917436827897632725596, −1.10972409181099362729520177762, 2.18978397533391814036747931945, 3.55687267388023006281141899658, 4.64125128726007515816578639374, 6.05417870303157348316457814810, 6.97801200137537074962552513522, 8.240273894397580355119995880381, 8.997781450349104837938677294321, 10.15732500362660617886802812460, 10.86694365316379292774536494004, 11.61140694244407094634268903152

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