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Properties

Label 2-425-25.16-c1-0-12
Degree $2$
Conductor $425$
Sign $-0.187 - 0.982i$
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  + (2.11 + 1.53i)2-s + (−0.309 − 0.951i)3-s + (1.5 + 4.61i)4-s + (−1.80 + 1.31i)5-s + (0.809 − 2.48i)6-s + 2.38·7-s + (−2.30 + 7.10i)8-s + (1.61 − 1.17i)9-s − 5.85·10-s + (1 + 0.726i)11-s + (3.92 − 2.85i)12-s + (−3.92 + 2.85i)13-s + (5.04 + 3.66i)14-s + (1.80 + 1.31i)15-s + (−7.97 + 5.79i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (1.49 + 1.08i)2-s + (−0.178 − 0.549i)3-s + (0.750 + 2.30i)4-s + (−0.809 + 0.587i)5-s + (0.330 − 1.01i)6-s + 0.900·7-s + (−0.816 + 2.51i)8-s + (0.539 − 0.391i)9-s − 1.85·10-s + (0.301 + 0.219i)11-s + (1.13 − 0.823i)12-s + (−1.08 + 0.791i)13-s + (1.34 + 0.979i)14-s + (0.467 + 0.339i)15-s + (−1.99 + 1.44i)16-s + (0.0749 − 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.187 - 0.982i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.187 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76966 + 2.13915i\)
\(L(\frac12)\) \(\approx\) \(1.76966 + 2.13915i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.80 - 1.31i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (-2.11 - 1.53i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.309 + 0.951i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 2.38T + 7T^{2} \)
11 \( 1 + (-1 - 0.726i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (3.92 - 2.85i)T + (4.01 - 12.3i)T^{2} \)
19 \( 1 + (0.572 - 1.76i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.80 - 1.31i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.66 + 8.19i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.78 + 5.48i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.42 + 4.66i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.23 + 3.80i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 0.145T + 43T^{2} \)
47 \( 1 + (-0.118 - 0.363i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.54 - 7.83i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.59 - 5.51i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.04 - 2.93i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-3.52 + 10.8i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.354 - 1.08i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.28 + 5.29i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.80 + 8.64i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.69 - 11.3i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-2.61 - 1.90i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.59 - 14.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.93683177493436095962805489151, −11.11596503216200290882976546342, −9.461227991021443216943959197987, −7.83513166018730250097943134539, −7.55660703827600231630049844876, −6.74137006495031240695977861753, −5.84160848854306944856703071346, −4.47685863451723831467101154878, −4.04382901026446601482372566799, −2.41893618931540249898134203754, 1.38580729821920268617424776165, 2.98284227316966226655030081373, 4.16491661022648277428382136723, 4.86828321957478443403010953717, 5.34755731956737623709459662831, 7.00395077080954744992830460867, 8.191232807707365442304670499309, 9.537694924828663209716336728405, 10.51343492619967840568366219189, 11.13648396465564339352862546304

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