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Properties

Label 2-315-63.4-c1-0-25
Degree $2$
Conductor $315$
Sign $-0.344 + 0.938i$
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  + (0.588 − 1.01i)2-s + (−0.114 − 1.72i)3-s + (0.306 + 0.530i)4-s + 5-s + (−1.83 − 0.900i)6-s + (−1.15 − 2.38i)7-s + 3.07·8-s + (−2.97 + 0.396i)9-s + (0.588 − 1.01i)10-s − 1.60·11-s + (0.882 − 0.590i)12-s + (2.48 − 4.30i)13-s + (−3.10 − 0.223i)14-s + (−0.114 − 1.72i)15-s + (1.19 − 2.07i)16-s + (0.876 − 1.51i)17-s + ⋯
L(s)  = 1  + (0.416 − 0.721i)2-s + (−0.0661 − 0.997i)3-s + (0.153 + 0.265i)4-s + 0.447·5-s + (−0.747 − 0.367i)6-s + (−0.436 − 0.899i)7-s + 1.08·8-s + (−0.991 + 0.132i)9-s + (0.186 − 0.322i)10-s − 0.484·11-s + (0.254 − 0.170i)12-s + (0.689 − 1.19i)13-s + (−0.830 − 0.0597i)14-s + (−0.0295 − 0.446i)15-s + (0.299 − 0.519i)16-s + (0.212 − 0.368i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.344 + 0.938i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.344 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.995634 - 1.42612i\)
\(L(\frac12)\) \(\approx\) \(0.995634 - 1.42612i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.114 + 1.72i)T \)
5 \( 1 - T \)
7 \( 1 + (1.15 + 2.38i)T \)
good2 \( 1 + (-0.588 + 1.01i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + 1.60T + 11T^{2} \)
13 \( 1 + (-2.48 + 4.30i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.876 + 1.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.57 - 6.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.47T + 23T^{2} \)
29 \( 1 + (1.18 + 2.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.01 - 3.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.22 - 7.31i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.20 + 3.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.404 + 0.699i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.23 - 7.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.44 + 2.50i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.69 - 11.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.44 + 9.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.05 - 7.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.85T + 71T^{2} \)
73 \( 1 + (-5.87 + 10.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.958 - 1.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.628 - 1.08i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.79 + 6.57i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.89 - 10.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.59575383033854313172908518041, −10.54031086855576045839306766887, −9.945415737957016621074973982223, −8.075262040729274395137509685967, −7.72037549868969795788787888877, −6.47066753745775565168035923386, −5.45280714391824686340832849882, −3.75440692055506465882690958449, −2.76780090074777496632378656099, −1.27961741365925856541876637268, 2.35205598847546223476994553243, 4.00753958534541956988071214345, 5.18818548069654452309587908411, 5.88874838177436610724608456030, 6.72068719061346403215401126321, 8.215835662262735484494934347525, 9.315992912610519496198014632421, 9.907973775485313476170079661699, 11.02578352256170466807625257984, 11.71819522229954141595491445348

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