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Properties

Label 2-425-17.13-c1-0-22
Degree $2$
Conductor $425$
Sign $-0.980 - 0.195i$
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  − 2.24i·2-s + (−0.140 − 0.140i)3-s − 3.05·4-s + (−0.314 + 0.314i)6-s + (1.33 − 1.33i)7-s + 2.37i·8-s − 2.96i·9-s + (2.49 − 2.49i)11-s + (0.428 + 0.428i)12-s − 1.27·13-s + (−3.00 − 3.00i)14-s − 0.766·16-s + (−3.68 + 1.85i)17-s − 6.65·18-s + 4.69i·19-s + ⋯
L(s)  = 1  − 1.59i·2-s + (−0.0808 − 0.0808i)3-s − 1.52·4-s + (−0.128 + 0.128i)6-s + (0.505 − 0.505i)7-s + 0.840i·8-s − 0.986i·9-s + (0.752 − 0.752i)11-s + (0.123 + 0.123i)12-s − 0.353·13-s + (−0.804 − 0.804i)14-s − 0.191·16-s + (−0.893 + 0.449i)17-s − 1.56·18-s + 1.07i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.980 - 0.195i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.980 - 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117128 + 1.18470i\)
\(L(\frac12)\) \(\approx\) \(0.117128 + 1.18470i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.68 - 1.85i)T \)
good2 \( 1 + 2.24iT - 2T^{2} \)
3 \( 1 + (0.140 + 0.140i)T + 3iT^{2} \)
7 \( 1 + (-1.33 + 1.33i)T - 7iT^{2} \)
11 \( 1 + (-2.49 + 2.49i)T - 11iT^{2} \)
13 \( 1 + 1.27T + 13T^{2} \)
19 \( 1 - 4.69iT - 19T^{2} \)
23 \( 1 + (-0.406 + 0.406i)T - 23iT^{2} \)
29 \( 1 + (3.81 + 3.81i)T + 29iT^{2} \)
31 \( 1 + (4.39 + 4.39i)T + 31iT^{2} \)
37 \( 1 + (-6.00 - 6.00i)T + 37iT^{2} \)
41 \( 1 + (-4.28 + 4.28i)T - 41iT^{2} \)
43 \( 1 + 9.16iT - 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 9.90iT - 53T^{2} \)
59 \( 1 + 3.15iT - 59T^{2} \)
61 \( 1 + (-3.63 + 3.63i)T - 61iT^{2} \)
67 \( 1 + 0.281T + 67T^{2} \)
71 \( 1 + (-8.30 - 8.30i)T + 71iT^{2} \)
73 \( 1 + (-6.95 - 6.95i)T + 73iT^{2} \)
79 \( 1 + (11.9 - 11.9i)T - 79iT^{2} \)
83 \( 1 - 8.51iT - 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + (4.18 + 4.18i)T + 97iT^{2} \)
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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

−10.99438954315923072193109146295, −9.936549197144014218885491980253, −9.232856264128688208229206004480, −8.302559525945363417129076976213, −6.93386910444461926109644220431, −5.78535271433831651990511736649, −4.15041568830681226717066443128, −3.69086205948017133142668399181, −2.13296026176283081126478299432, −0.798235789122958250204344552123, 2.26689207800569984707263466951, 4.47618839874085536885677028207, 5.02002627970418897134385322332, 6.07050324052252353786254843697, 7.18239728740339391756197923784, 7.63314406195769170991243916296, 8.908156428993966611439783299321, 9.270733340205235556274746075159, 10.80371933537512139980493607472, 11.56081921728509270509624029356

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