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Properties

Label 2-15-5.3-c4-0-0
Degree $2$
Conductor $15$
Sign $-0.862 + 0.505i$
Analytic cond. $1.55054$
Root an. cond. $1.24521$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.02 + 5.02i)2-s + (−3.67 − 3.67i)3-s − 34.5i·4-s + (−23.8 + 7.35i)5-s + 36.9·6-s + (−38.5 + 38.5i)7-s + (93.3 + 93.3i)8-s + 27i·9-s + (83.1 − 157. i)10-s − 40.4·11-s + (−126. + 126. i)12-s + (20.8 + 20.8i)13-s − 387. i·14-s + (114. + 60.7i)15-s − 385.·16-s + (15.8 − 15.8i)17-s + ⋯
L(s)  = 1  + (−1.25 + 1.25i)2-s + (−0.408 − 0.408i)3-s − 2.16i·4-s + (−0.955 + 0.294i)5-s + 1.02·6-s + (−0.786 + 0.786i)7-s + (1.45 + 1.45i)8-s + 0.333i·9-s + (0.831 − 1.57i)10-s − 0.334·11-s + (−0.881 + 0.881i)12-s + (0.123 + 0.123i)13-s − 1.97i·14-s + (0.510 + 0.270i)15-s − 1.50·16-s + (0.0549 − 0.0549i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 + 0.505i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.862 + 0.505i$
Analytic conductor: \(1.55054\)
Root analytic conductor: \(1.24521\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :2),\ -0.862 + 0.505i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0391874 - 0.144324i\)
\(L(\frac12)\) \(\approx\) \(0.0391874 - 0.144324i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.67 + 3.67i)T \)
5 \( 1 + (23.8 - 7.35i)T \)
good2 \( 1 + (5.02 - 5.02i)T - 16iT^{2} \)
7 \( 1 + (38.5 - 38.5i)T - 2.40e3iT^{2} \)
11 \( 1 + 40.4T + 1.46e4T^{2} \)
13 \( 1 + (-20.8 - 20.8i)T + 2.85e4iT^{2} \)
17 \( 1 + (-15.8 + 15.8i)T - 8.35e4iT^{2} \)
19 \( 1 - 314. iT - 1.30e5T^{2} \)
23 \( 1 + (572. + 572. i)T + 2.79e5iT^{2} \)
29 \( 1 - 824. iT - 7.07e5T^{2} \)
31 \( 1 + 1.34e3T + 9.23e5T^{2} \)
37 \( 1 + (589. - 589. i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.85e3T + 2.82e6T^{2} \)
43 \( 1 + (671. + 671. i)T + 3.41e6iT^{2} \)
47 \( 1 + (-504. + 504. i)T - 4.87e6iT^{2} \)
53 \( 1 + (-2.25e3 - 2.25e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 2.58e3iT - 1.21e7T^{2} \)
61 \( 1 + 3.27e3T + 1.38e7T^{2} \)
67 \( 1 + (3.42e3 - 3.42e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 5.67e3T + 2.54e7T^{2} \)
73 \( 1 + (-4.45e3 - 4.45e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 6.46e3iT - 3.89e7T^{2} \)
83 \( 1 + (621. + 621. i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.85e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.23e4 + 1.23e4i)T - 8.85e7iT^{2} \)
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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

−18.71057542954056063293633360730, −18.28160474163088767640439660194, −16.53374352298555029148243189394, −15.88982514500399706270247930820, −14.62952164479728975525094164095, −12.30730083002609884674798921612, −10.46426520947176559019005025054, −8.724672166095888282844361314084, −7.37330813500084200737981940299, −5.99442174008626375324395730642, 0.22968909003146704177991085435, 3.71299286844569129789295370472, 7.62942490501046037941408650844, 9.327751814043634625098753084410, 10.60102867045620915179571446598, 11.68425161448185048660060215695, 13.00018857370725650164531414767, 15.80451969678199961457129648502, 16.79198657802066971530468240140, 18.02067714797115870253188105292

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