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Properties

Label 2-15-15.2-c5-0-0
Degree $2$
Conductor $15$
Sign $-0.801 - 0.598i$
Analytic cond. $2.40575$
Root an. cond. $1.55105$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.879 + 0.879i)2-s + (−15.5 − 0.0935i)3-s + 30.4i·4-s + (−51.8 + 20.8i)5-s + (13.7 − 13.6i)6-s + (−11.3 − 11.3i)7-s + (−54.9 − 54.9i)8-s + (242. + 2.91i)9-s + (27.3 − 63.9i)10-s + 359. i·11-s + (2.84 − 474. i)12-s + (370. − 370. i)13-s + 19.9·14-s + (810. − 319. i)15-s − 877.·16-s + (−1.24e3 + 1.24e3i)17-s + ⋯
L(s)  = 1  + (−0.155 + 0.155i)2-s + (−0.999 − 0.00600i)3-s + 0.951i·4-s + (−0.928 + 0.372i)5-s + (0.156 − 0.154i)6-s + (−0.0874 − 0.0874i)7-s + (−0.303 − 0.303i)8-s + (0.999 + 0.0120i)9-s + (0.0864 − 0.202i)10-s + 0.896i·11-s + (0.00571 − 0.951i)12-s + (0.607 − 0.607i)13-s + 0.0272·14-s + (0.930 − 0.366i)15-s − 0.857·16-s + (−1.04 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.801 - 0.598i$
Analytic conductor: \(2.40575\)
Root analytic conductor: \(1.55105\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5/2),\ -0.801 - 0.598i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.167514 + 0.503979i\)
\(L(\frac12)\) \(\approx\) \(0.167514 + 0.503979i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (15.5 + 0.0935i)T \)
5 \( 1 + (51.8 - 20.8i)T \)
good2 \( 1 + (0.879 - 0.879i)T - 32iT^{2} \)
7 \( 1 + (11.3 + 11.3i)T + 1.68e4iT^{2} \)
11 \( 1 - 359. iT - 1.61e5T^{2} \)
13 \( 1 + (-370. + 370. i)T - 3.71e5iT^{2} \)
17 \( 1 + (1.24e3 - 1.24e3i)T - 1.41e6iT^{2} \)
19 \( 1 - 2.00e3iT - 2.47e6T^{2} \)
23 \( 1 + (245. + 245. i)T + 6.43e6iT^{2} \)
29 \( 1 - 7.05e3T + 2.05e7T^{2} \)
31 \( 1 + 3.03e3T + 2.86e7T^{2} \)
37 \( 1 + (-2.36e3 - 2.36e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 6.43e3iT - 1.15e8T^{2} \)
43 \( 1 + (9.71e3 - 9.71e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (1.49e3 - 1.49e3i)T - 2.29e8iT^{2} \)
53 \( 1 + (-1.87e4 - 1.87e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 2.12e4T + 7.14e8T^{2} \)
61 \( 1 + 691.T + 8.44e8T^{2} \)
67 \( 1 + (-4.05e4 - 4.05e4i)T + 1.35e9iT^{2} \)
71 \( 1 - 8.44e3iT - 1.80e9T^{2} \)
73 \( 1 + (1.16e4 - 1.16e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 2.98e4iT - 3.07e9T^{2} \)
83 \( 1 + (-3.06e4 - 3.06e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 1.06e5T + 5.58e9T^{2} \)
97 \( 1 + (3.31e4 + 3.31e4i)T + 8.58e9iT^{2} \)
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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

−18.43572585283257965235185949304, −17.51056295141656082776397216313, −16.27356351526325809735644011902, −15.28270785697160948287131829794, −12.86463002197611792616586580518, −11.91950638669645236798116503634, −10.51479122532084467998627333250, −8.118439301755867476030298499460, −6.68395585833447635949171037711, −4.07155529384468121421401844825, 0.53834597883874541952000279013, 4.82671176095074523106285874661, 6.61833731561102045249896667770, 9.037178237645944389802601770136, 10.93496023098910657612642966998, 11.67475554721010373131882041844, 13.54973977056844404115116623372, 15.51441297541790330912536519983, 16.25394959177529827159804732305, 17.98186160059817464962634485264

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