↑

Properties

Label 2-315-105.2-c1-0-10
Degree $2$
Conductor $315$
Sign $0.116 + 0.993i$
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  + (−0.271 − 1.01i)2-s + (0.777 − 0.449i)4-s + (0.310 + 2.21i)5-s + (−1.26 − 2.32i)7-s + (−2.15 − 2.15i)8-s + (2.16 − 0.915i)10-s + (1.52 − 0.880i)11-s + (4.98 − 4.98i)13-s + (−2.00 + 1.91i)14-s + (−0.698 + 1.20i)16-s + (−0.458 − 0.122i)17-s + (2.33 + 1.34i)19-s + (1.23 + 1.58i)20-s + (−1.30 − 1.30i)22-s + (6.30 − 1.69i)23-s + ⋯
L(s)  = 1  + (−0.192 − 0.716i)2-s + (0.388 − 0.224i)4-s + (0.138 + 0.990i)5-s + (−0.479 − 0.877i)7-s + (−0.760 − 0.760i)8-s + (0.683 − 0.289i)10-s + (0.460 − 0.265i)11-s + (1.38 − 1.38i)13-s + (−0.537 + 0.512i)14-s + (−0.174 + 0.302i)16-s + (−0.111 − 0.0298i)17-s + (0.535 + 0.309i)19-s + (0.276 + 0.354i)20-s + (−0.278 − 0.278i)22-s + (1.31 − 0.352i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.116 + 0.993i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.116 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00236 - 0.891652i\)
\(L(\frac12)\) \(\approx\) \(1.00236 - 0.891652i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.310 - 2.21i)T \)
7 \( 1 + (1.26 + 2.32i)T \)
good2 \( 1 + (0.271 + 1.01i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-1.52 + 0.880i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.98 + 4.98i)T - 13iT^{2} \)
17 \( 1 + (0.458 + 0.122i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.33 - 1.34i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.30 + 1.69i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 7.01T + 29T^{2} \)
31 \( 1 + (-3.95 - 6.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.45 - 1.73i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.20iT - 41T^{2} \)
43 \( 1 + (2.28 - 2.28i)T - 43iT^{2} \)
47 \( 1 + (-0.393 - 1.47i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.26 - 8.45i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-6.17 - 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.26 - 3.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.530 - 1.98i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.79iT - 71T^{2} \)
73 \( 1 + (1.45 + 0.391i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-9.08 - 5.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.07 - 2.07i)T + 83iT^{2} \)
89 \( 1 + (1.64 - 2.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.21 - 3.21i)T + 97iT^{2} \)
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

−10.98891973598577154168377713234, −10.73964761436443235769481057262, −9.979168038287854244614818926243, −8.850831785982318317912503265358, −7.40935926710054330592941424063, −6.60532091253162385557262733208, −5.71874581594749194568515617885, −3.60122329924608024647323641637, −3.04483942163514059628854231155, −1.16628487039200935812765593635, 1.89597197117038354513015934715, 3.61657265759554096277690878136, 5.14471999768315894877327339257, 6.14572409182681565000113345903, 6.90261446601307368916664607749, 8.206703024199377338571576118864, 9.020849616006131474583868020706, 9.452856773281140720889864223872, 11.37609188272756020703578260650, 11.69174642420970983274282278623

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