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Properties

Label 2-425-17.4-c1-0-6
Degree $2$
Conductor $425$
Sign $-0.688 - 0.725i$
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  + 1.80i·2-s + (0.397 − 0.397i)3-s − 1.25·4-s + (0.716 + 0.716i)6-s + (−2.20 − 2.20i)7-s + 1.34i·8-s + 2.68i·9-s + (3.96 + 3.96i)11-s + (−0.497 + 0.497i)12-s − 1.24·13-s + (3.96 − 3.96i)14-s − 4.93·16-s + (0.397 + 4.10i)17-s − 4.84·18-s + 4i·19-s + ⋯
L(s)  = 1  + 1.27i·2-s + (0.229 − 0.229i)3-s − 0.626·4-s + (0.292 + 0.292i)6-s + (−0.831 − 0.831i)7-s + 0.476i·8-s + 0.894i·9-s + (1.19 + 1.19i)11-s + (−0.143 + 0.143i)12-s − 0.346·13-s + (1.06 − 1.06i)14-s − 1.23·16-s + (0.0963 + 0.995i)17-s − 1.14·18-s + 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.688 - 0.725i$
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: \((2,\ 425,\ (\ :1/2),\ -0.688 - 0.725i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.552204 + 1.28559i\)
\(L(\frac12)\) \(\approx\) \(0.552204 + 1.28559i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.397 - 4.10i)T \)
good2 \( 1 - 1.80iT - 2T^{2} \)
3 \( 1 + (-0.397 + 0.397i)T - 3iT^{2} \)
7 \( 1 + (2.20 + 2.20i)T + 7iT^{2} \)
11 \( 1 + (-3.96 - 3.96i)T + 11iT^{2} \)
13 \( 1 + 1.24T + 13T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-1.64 - 1.64i)T + 23iT^{2} \)
29 \( 1 + (-4.68 + 4.68i)T - 29iT^{2} \)
31 \( 1 + (-3.22 + 3.22i)T - 31iT^{2} \)
37 \( 1 + (-1.34 + 1.34i)T - 37iT^{2} \)
41 \( 1 + (4.18 + 4.18i)T + 41iT^{2} \)
43 \( 1 + 2.04iT - 43T^{2} \)
47 \( 1 + 4.85T + 47T^{2} \)
53 \( 1 + 9.11iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + (4 + 4i)T + 61iT^{2} \)
67 \( 1 - 8.46T + 67T^{2} \)
71 \( 1 + (6.22 - 6.22i)T - 71iT^{2} \)
73 \( 1 + (-1.10 + 1.10i)T - 73iT^{2} \)
79 \( 1 + (-3.47 - 3.47i)T + 79iT^{2} \)
83 \( 1 - 3.94iT - 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + (-8.76 + 8.76i)T - 97iT^{2} \)
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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

−11.58108787224652412594780927855, −10.32220877760501341631529408846, −9.653147509327756667720524826456, −8.407075850779174198990963657034, −7.64441000430027708307117628571, −6.86796655642902616159087780090, −6.22597865082616807483704622761, −4.89974349506535155716533949123, −3.83833413607463902120938142761, −1.98528786946884140862143848008, 0.918298451738727588236256904232, 2.89347020077624276061313384359, 3.24244236062326481042798620946, 4.61849093864554916779094385604, 6.22890085914695263271204872131, 6.84153473770993894298960140641, 8.786259621711948093896280330559, 9.162426128423418112350887033608, 9.902492968224351195605945165729, 10.98121139139584670414442055488

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