↑

Properties

Label 2-425-85.9-c1-0-23
Degree $2$
Conductor $425$
Sign $-0.932 - 0.361i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
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  + (−1.09 − 1.09i)2-s + (1.15 − 2.77i)3-s + 0.419i·4-s + (−4.32 + 1.79i)6-s + (3.19 − 1.32i)7-s + (−1.73 + 1.73i)8-s + (−4.27 − 4.27i)9-s + (−3.92 + 1.62i)11-s + (1.16 + 0.483i)12-s − 0.127·13-s + (−4.96 − 2.05i)14-s + 4.66·16-s + (−0.193 − 4.11i)17-s + 9.40i·18-s + (−1.81 + 1.81i)19-s + ⋯
L(s)  = 1  + (−0.777 − 0.777i)2-s + (0.664 − 1.60i)3-s + 0.209i·4-s + (−1.76 + 0.731i)6-s + (1.20 − 0.499i)7-s + (−0.614 + 0.614i)8-s + (−1.42 − 1.42i)9-s + (−1.18 + 0.489i)11-s + (0.336 + 0.139i)12-s − 0.0353·13-s + (−1.32 − 0.549i)14-s + 1.16·16-s + (−0.0468 − 0.998i)17-s + 2.21i·18-s + (−0.417 + 0.417i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.932 - 0.361i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.932 - 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192840 + 1.02982i\)
\(L(\frac12)\) \(\approx\) \(0.192840 + 1.02982i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (0.193 + 4.11i)T \)
good2 \( 1 + (1.09 + 1.09i)T + 2iT^{2} \)
3 \( 1 + (-1.15 + 2.77i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-3.19 + 1.32i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (3.92 - 1.62i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 0.127T + 13T^{2} \)
19 \( 1 + (1.81 - 1.81i)T - 19iT^{2} \)
23 \( 1 + (1.24 + 3.00i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.87 + 4.53i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-4.95 - 2.05i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-0.677 + 1.63i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.85 - 9.29i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.79 + 1.79i)T - 43iT^{2} \)
47 \( 1 - 4.59T + 47T^{2} \)
53 \( 1 + (1.15 + 1.15i)T + 53iT^{2} \)
59 \( 1 + (-4.34 - 4.34i)T + 59iT^{2} \)
61 \( 1 + (1.54 + 3.73i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 6.88iT - 67T^{2} \)
71 \( 1 + (6.66 + 2.76i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-13.5 - 5.59i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-4.75 + 1.97i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (10.2 + 10.2i)T + 83iT^{2} \)
89 \( 1 + 0.600iT - 89T^{2} \)
97 \( 1 + (7.09 + 2.93i)T + (68.5 + 68.5i)T^{2} \)
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

−10.72165614724571760441507174360, −9.807322020468836258300810194780, −8.619445994567603270967169337487, −7.992536177841708358315359130086, −7.43035287914925223245213302663, −6.15929795491792780342042649173, −4.79360280079853688664152900954, −2.73407162774005623080993699211, −2.01495593976150707968472228175, −0.805226164029447342408048202343, 2.60396688564268254126765248727, 3.85693480139781059712611894757, 4.98032907942459771054052194517, 5.86746510494510623079091691046, 7.54345283245925969893886555870, 8.470108073895258219067017827894, 8.586790052982706931387120209836, 9.696668012164997491222619613709, 10.53433946121537585826566608132, 11.20253726601645214801060318943

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