↑

Properties

Label 2-966-69.68-c1-0-4
Degree $2$
Conductor $966$
Sign $-0.727 - 0.685i$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
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  + i·2-s + (0.696 − 1.58i)3-s − 4-s − 3.16·5-s + (1.58 + 0.696i)6-s − i·7-s − i·8-s + (−2.02 − 2.20i)9-s − 3.16i·10-s − 1.43·11-s + (−0.696 + 1.58i)12-s + 2.73·13-s + 14-s + (−2.20 + 5.01i)15-s + 16-s − 0.955·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.402 − 0.915i)3-s − 0.5·4-s − 1.41·5-s + (0.647 + 0.284i)6-s − 0.377i·7-s − 0.353i·8-s + (−0.676 − 0.736i)9-s − 0.999i·10-s − 0.433·11-s + (−0.201 + 0.457i)12-s + 0.759·13-s + 0.267·14-s + (−0.568 + 1.29i)15-s + 0.250·16-s − 0.231·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.727 - 0.685i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.727 - 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.157007 + 0.395640i\)
\(L(\frac12)\) \(\approx\) \(0.157007 + 0.395640i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.696 + 1.58i)T \)
7 \( 1 + iT \)
23 \( 1 + (1.60 - 4.51i)T \)
good5 \( 1 + 3.16T + 5T^{2} \)
11 \( 1 + 1.43T + 11T^{2} \)
13 \( 1 - 2.73T + 13T^{2} \)
17 \( 1 + 0.955T + 17T^{2} \)
19 \( 1 - 7.49iT - 19T^{2} \)
29 \( 1 - 8.82iT - 29T^{2} \)
31 \( 1 + 2.94T + 31T^{2} \)
37 \( 1 - 2.44iT - 37T^{2} \)
41 \( 1 + 3.05iT - 41T^{2} \)
43 \( 1 - 2.41iT - 43T^{2} \)
47 \( 1 + 8.46iT - 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 + 4.28iT - 59T^{2} \)
61 \( 1 - 3.13iT - 61T^{2} \)
67 \( 1 + 2.67iT - 67T^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + 4.14T + 73T^{2} \)
79 \( 1 + 4.98iT - 79T^{2} \)
83 \( 1 + 5.69T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 - 4.71iT - 97T^{2} \)
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

−10.33694704508940199699992884819, −9.101870350536444449320720027525, −8.275850164948886207759233662148, −7.81616132340744020293707934876, −7.18828072008365319658594473528, −6.30307719900370751343550135673, −5.27824660699511788990363206044, −3.87241047712412117816889697784, −3.38381376664548373533987707807, −1.44854045134940720279711950831, 0.19513025722752203008878120101, 2.45516251657205827543461199990, 3.31527113213068454756341631315, 4.28143116878127849588549253068, 4.78777187663751040337744034357, 6.11107539894033387620288582247, 7.54815592529873816303251775650, 8.278748424155600908718165394029, 8.908279593142014183343564809880, 9.687009342687849457684217228309

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