↑

Properties

Label 2-315-63.58-c1-0-19
Degree $2$
Conductor $315$
Sign $0.998 - 0.0467i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.32·2-s + (−1.66 − 0.462i)3-s + 3.40·4-s + (0.5 + 0.866i)5-s + (−3.88 − 1.07i)6-s + (2.02 + 1.70i)7-s + 3.27·8-s + (2.57 + 1.54i)9-s + (1.16 + 2.01i)10-s + (1.47 − 2.56i)11-s + (−5.68 − 1.57i)12-s + (1.00 − 1.73i)13-s + (4.71 + 3.95i)14-s + (−0.434 − 1.67i)15-s + 0.799·16-s + (−1.98 − 3.43i)17-s + ⋯
L(s)  = 1  + 1.64·2-s + (−0.963 − 0.266i)3-s + 1.70·4-s + (0.223 + 0.387i)5-s + (−1.58 − 0.438i)6-s + (0.766 + 0.642i)7-s + 1.15·8-s + (0.857 + 0.514i)9-s + (0.367 + 0.636i)10-s + (0.445 − 0.772i)11-s + (−1.64 − 0.454i)12-s + (0.278 − 0.482i)13-s + (1.25 + 1.05i)14-s + (−0.112 − 0.432i)15-s + 0.199·16-s + (−0.480 − 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.998 - 0.0467i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.998 - 0.0467i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.58091 + 0.0603122i\)
\(L(\frac12)\) \(\approx\) \(2.58091 + 0.0603122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.66 + 0.462i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.02 - 1.70i)T \)
good2 \( 1 - 2.32T + 2T^{2} \)
11 \( 1 + (-1.47 + 2.56i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.00 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.98 + 3.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.55 - 4.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.216 + 0.375i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.68 - 2.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.06T + 31T^{2} \)
37 \( 1 + (-0.0400 + 0.0693i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.435 - 0.753i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.02 + 1.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.84T + 47T^{2} \)
53 \( 1 + (4.24 + 7.35i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 15.0T + 61T^{2} \)
67 \( 1 + 9.09T + 67T^{2} \)
71 \( 1 + 2.95T + 71T^{2} \)
73 \( 1 + (5.84 + 10.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + (0.126 + 0.218i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.58 + 14.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.67 - 13.2i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.63312513707605515283809128222, −11.33610134157708554429568130860, −10.35487787187361387623262204373, −8.717318132912460806546945128015, −7.32238665337980444549845330393, −6.28656160986397473918479875960, −5.63990689491300386115542383386, −4.82741410929448607275564418514, −3.54226760939269210545549995296, −2.00164757017377973945560341029, 1.82724369097947077447647887003, 4.01324298016004244649750381645, 4.48498991556591572916130602674, 5.41834876737923611514040949699, 6.46590554189728977812006269432, 7.20930050050420936282222417950, 8.904056009609693654706516526862, 10.22114587285504352096393722925, 11.22351983789501804785464761289, 11.64694178228502322112896757028

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