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Properties

Label 2-315-63.5-c1-0-17
Degree $2$
Conductor $315$
Sign $0.868 - 0.495i$
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  + 0.524i·2-s + (1.63 + 0.571i)3-s + 1.72·4-s + (−0.5 − 0.866i)5-s + (−0.299 + 0.857i)6-s + (−1.53 − 2.15i)7-s + 1.95i·8-s + (2.34 + 1.87i)9-s + (0.454 − 0.262i)10-s + (−0.593 − 0.342i)11-s + (2.82 + 0.986i)12-s + (5.48 + 3.16i)13-s + (1.12 − 0.806i)14-s + (−0.322 − 1.70i)15-s + 2.42·16-s + (−1.00 − 1.74i)17-s + ⋯
L(s)  = 1  + 0.370i·2-s + (0.943 + 0.330i)3-s + 0.862·4-s + (−0.223 − 0.387i)5-s + (−0.122 + 0.349i)6-s + (−0.581 − 0.813i)7-s + 0.690i·8-s + (0.781 + 0.623i)9-s + (0.143 − 0.0829i)10-s + (−0.178 − 0.103i)11-s + (0.814 + 0.284i)12-s + (1.52 + 0.877i)13-s + (0.301 − 0.215i)14-s + (−0.0831 − 0.439i)15-s + 0.606·16-s + (−0.244 − 0.423i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.868 - 0.495i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.868 - 0.495i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90626 + 0.505938i\)
\(L(\frac12)\) \(\approx\) \(1.90626 + 0.505938i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.63 - 0.571i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (1.53 + 2.15i)T \)
good2 \( 1 - 0.524iT - 2T^{2} \)
11 \( 1 + (0.593 + 0.342i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.48 - 3.16i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.00 + 1.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.11 + 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.98 - 4.03i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.246 + 0.142i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.59iT - 31T^{2} \)
37 \( 1 + (0.593 - 1.02i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.16 + 5.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.53 + 2.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + (8.01 - 4.62i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.27T + 59T^{2} \)
61 \( 1 + 1.96iT - 61T^{2} \)
67 \( 1 - 4.13T + 67T^{2} \)
71 \( 1 + 3.56iT - 71T^{2} \)
73 \( 1 + (3.24 - 1.87i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 4.85T + 79T^{2} \)
83 \( 1 + (2.12 + 3.68i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.97 + 5.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.55 - 4.93i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.53293839274094011492226101574, −10.81112887294778939469647768066, −9.802130995773689000345154425583, −8.755935695175183050791757976139, −7.951471226569717279405664153706, −6.98662102881262714641408291910, −6.08458888096185728005626218934, −4.37622082246862505000824163957, −3.45249148625233376155309978055, −1.91469053184501773788932103928, 1.86444152343744518300331618465, 2.98153611317732605433191502677, 3.83995200346919864389050029320, 6.12609363434605256592384209798, 6.55604010651823766068825173395, 8.035627932150053640493794646387, 8.496294131495938626331188964148, 9.914238619776892794094578005491, 10.57192671719926776878630938540, 11.62943140185906253178384822324

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