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Properties

Label 2-315-21.5-c1-0-5
Degree $2$
Conductor $315$
Sign $-0.145 + 0.989i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  + (−0.986 − 0.569i)2-s + (−0.350 − 0.607i)4-s + (−0.5 + 0.866i)5-s + (2.18 − 1.48i)7-s + 3.07i·8-s + (0.986 − 0.569i)10-s + (2.09 − 1.21i)11-s − 3.64i·13-s + (−3.00 + 0.218i)14-s + (1.05 − 1.82i)16-s + (−2.74 − 4.75i)17-s + (0.387 + 0.223i)19-s + 0.701·20-s − 2.76·22-s + (−3.51 − 2.02i)23-s + ⋯
L(s)  = 1  + (−0.697 − 0.402i)2-s + (−0.175 − 0.303i)4-s + (−0.223 + 0.387i)5-s + (0.827 − 0.561i)7-s + 1.08i·8-s + (0.312 − 0.180i)10-s + (0.632 − 0.365i)11-s − 1.01i·13-s + (−0.803 + 0.0584i)14-s + (0.263 − 0.455i)16-s + (−0.665 − 1.15i)17-s + (0.0889 + 0.0513i)19-s + 0.156·20-s − 0.588·22-s + (−0.732 − 0.423i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.145 + 0.989i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.145 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534912 - 0.619345i\)
\(L(\frac12)\) \(\approx\) \(0.534912 - 0.619345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.18 + 1.48i)T \)
good2 \( 1 + (0.986 + 0.569i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-2.09 + 1.21i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.64iT - 13T^{2} \)
17 \( 1 + (2.74 + 4.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.387 - 0.223i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.51 + 2.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.40iT - 29T^{2} \)
31 \( 1 + (-7.70 + 4.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.42 - 7.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.14T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + (-0.908 + 1.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.38 + 1.95i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.00727 - 0.0126i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.19 - 5.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.27 - 7.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (-4.88 + 2.81i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.91 - 5.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.51T + 83T^{2} \)
89 \( 1 + (7.06 - 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.65iT - 97T^{2} \)
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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

−11.41790777403989256347023901517, −10.35406920885852105076208676278, −9.785005495037285194387344936151, −8.507831790529647916445678591547, −7.927349645920464283507762975183, −6.63719861842144224051144279822, −5.33884057578753646592509928675, −4.21023308956496750060978896036, −2.51377934380092110552034345965, −0.810393157021652804444693309800, 1.71922891199266515324985713402, 3.83458590407039347543858751173, 4.76583180843415351976352263962, 6.30045812001117254772501074381, 7.28437202314328654508789934266, 8.381768851008043813167101275136, 8.819377461491075394655257724379, 9.728884858963856406282554656998, 10.98531536578594805540636148065, 12.02510370187831347748593495048

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