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Properties

Label 2-425-17.4-c1-0-21
Degree $2$
Conductor $425$
Sign $-0.163 + 0.986i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.12i·2-s + (1.75 − 1.75i)3-s + 0.729·4-s + (−1.97 − 1.97i)6-s + (1.72 + 1.72i)7-s − 3.07i·8-s − 3.16i·9-s + (2.57 + 2.57i)11-s + (1.28 − 1.28i)12-s − 3.64·13-s + (1.94 − 1.94i)14-s − 2.00·16-s + (−3.03 + 2.79i)17-s − 3.56·18-s − 2.61i·19-s + ⋯
L(s)  = 1  − 0.796i·2-s + (1.01 − 1.01i)3-s + 0.364·4-s + (−0.807 − 0.807i)6-s + (0.652 + 0.652i)7-s − 1.08i·8-s − 1.05i·9-s + (0.775 + 0.775i)11-s + (0.369 − 0.369i)12-s − 1.01·13-s + (0.520 − 0.520i)14-s − 0.502·16-s + (−0.735 + 0.677i)17-s − 0.840·18-s − 0.599i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.163 + 0.986i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (276, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.163 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45158 - 1.71154i\)
\(L(\frac12)\) \(\approx\) \(1.45158 - 1.71154i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.03 - 2.79i)T \)
good2 \( 1 + 1.12iT - 2T^{2} \)
3 \( 1 + (-1.75 + 1.75i)T - 3iT^{2} \)
7 \( 1 + (-1.72 - 1.72i)T + 7iT^{2} \)
11 \( 1 + (-2.57 - 2.57i)T + 11iT^{2} \)
13 \( 1 + 3.64T + 13T^{2} \)
19 \( 1 + 2.61iT - 19T^{2} \)
23 \( 1 + (0.993 + 0.993i)T + 23iT^{2} \)
29 \( 1 + (-0.601 + 0.601i)T - 29iT^{2} \)
31 \( 1 + (6.67 - 6.67i)T - 31iT^{2} \)
37 \( 1 + (7.78 - 7.78i)T - 37iT^{2} \)
41 \( 1 + (6.74 + 6.74i)T + 41iT^{2} \)
43 \( 1 - 7.47iT - 43T^{2} \)
47 \( 1 - 5.42T + 47T^{2} \)
53 \( 1 + 12.9iT - 53T^{2} \)
59 \( 1 + 1.40iT - 59T^{2} \)
61 \( 1 + (-0.804 - 0.804i)T + 61iT^{2} \)
67 \( 1 - 2.07T + 67T^{2} \)
71 \( 1 + (-8.69 + 8.69i)T - 71iT^{2} \)
73 \( 1 + (1.04 - 1.04i)T - 73iT^{2} \)
79 \( 1 + (-6.34 - 6.34i)T + 79iT^{2} \)
83 \( 1 - 2.52iT - 83T^{2} \)
89 \( 1 + 1.66T + 89T^{2} \)
97 \( 1 + (-8.67 + 8.67i)T - 97iT^{2} \)
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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

−11.11086580201154877369663758249, −10.03631504435201476784758929409, −9.041781744476266884592524156566, −8.288952422963882346383890095021, −7.11850080964852136179793914460, −6.69252292122169124981354723956, −4.92446695177087950804107705087, −3.45882879475510456915603280400, −2.21478924343758568121041337397, −1.73346879741093572899547184143, 2.20640820135803467913573115292, 3.57083841786235917877760601424, 4.57783525924329251154697046707, 5.68136187895260904094996532467, 7.01428536248513362066713269782, 7.73836521528038261026195420245, 8.675582488743224808106050879294, 9.377047711393019468410169404749, 10.47668718812301426412672278227, 11.20950500020945467459196700325

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