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Properties

Label 2-315-63.16-c1-0-0
Degree $2$
Conductor $315$
Sign $-0.433 - 0.901i$
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  + (−0.259 − 0.449i)2-s + (−1.70 + 0.278i)3-s + (0.865 − 1.49i)4-s − 5-s + (0.568 + 0.695i)6-s + (−1.91 + 1.82i)7-s − 1.93·8-s + (2.84 − 0.952i)9-s + (0.259 + 0.449i)10-s − 4.16·11-s + (−1.06 + 2.80i)12-s + (2.27 + 3.93i)13-s + (1.31 + 0.388i)14-s + (1.70 − 0.278i)15-s + (−1.22 − 2.12i)16-s + (1.39 + 2.41i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.317i)2-s + (−0.986 + 0.160i)3-s + (0.432 − 0.749i)4-s − 0.447·5-s + (0.232 + 0.283i)6-s + (−0.724 + 0.688i)7-s − 0.683·8-s + (0.948 − 0.317i)9-s + (0.0819 + 0.141i)10-s − 1.25·11-s + (−0.306 + 0.809i)12-s + (0.629 + 1.09i)13-s + (0.351 + 0.103i)14-s + (0.441 − 0.0719i)15-s + (−0.307 − 0.532i)16-s + (0.338 + 0.586i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.433 - 0.901i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.433 - 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.132489 + 0.210716i\)
\(L(\frac12)\) \(\approx\) \(0.132489 + 0.210716i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.278i)T \)
5 \( 1 + T \)
7 \( 1 + (1.91 - 1.82i)T \)
good2 \( 1 + (0.259 + 0.449i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + 4.16T + 11T^{2} \)
13 \( 1 + (-2.27 - 3.93i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.39 - 2.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.22 - 3.85i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.45T + 23T^{2} \)
29 \( 1 + (2.18 - 3.77i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.859 + 1.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.66 - 8.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.217 - 0.376i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.19 + 8.99i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.75 + 6.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.17 + 7.22i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.37 - 9.30i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.05 + 3.55i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.77 + 6.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 + (1.11 + 1.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.495 + 0.858i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.763 + 1.32i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.04 + 1.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.43 - 14.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.96806317672398033882393209095, −10.91423649063461973959246899885, −10.31907042540557451329416011995, −9.498373349879881358970966081792, −8.250634983059099291868716724231, −6.78911571360993531643953633207, −6.04935959318845980152845148881, −5.21043560926409917161637098964, −3.68129549644845541788479368386, −1.91472137266163498145048718761, 0.20049248566334511044180527608, 2.87681696076173016013213328312, 4.16711589242641419286694534588, 5.61198636976748570230626446813, 6.56278858866594172827256509185, 7.55521100743540501070921478367, 8.041823264537880086035351428092, 9.626898048696087421824734622992, 10.74001230321308228918066091019, 11.17599434699388026510425858798

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