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Properties

Label 2-425-85.19-c1-0-0
Degree $2$
Conductor $425$
Sign $-0.552 - 0.833i$
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  + (−1.52 + 1.52i)2-s + (−1.06 − 2.56i)3-s − 2.67i·4-s + (5.54 + 2.29i)6-s + (−2.90 − 1.20i)7-s + (1.03 + 1.03i)8-s + (−3.32 + 3.32i)9-s + (−4.70 − 1.94i)11-s + (−6.86 + 2.84i)12-s + 1.39·13-s + (6.27 − 2.60i)14-s + 2.18·16-s + (3.66 + 1.88i)17-s − 10.1i·18-s + (3.63 + 3.63i)19-s + ⋯
L(s)  = 1  + (−1.08 + 1.08i)2-s + (−0.613 − 1.48i)3-s − 1.33i·4-s + (2.26 + 0.937i)6-s + (−1.09 − 0.454i)7-s + (0.366 + 0.366i)8-s + (−1.10 + 1.10i)9-s + (−1.41 − 0.587i)11-s + (−1.98 + 0.820i)12-s + 0.386·13-s + (1.67 − 0.695i)14-s + 0.545·16-s + (0.889 + 0.457i)17-s − 2.39i·18-s + (0.833 + 0.833i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.552 - 0.833i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.552 - 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0682320 + 0.127098i\)
\(L(\frac12)\) \(\approx\) \(0.0682320 + 0.127098i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.66 - 1.88i)T \)
good2 \( 1 + (1.52 - 1.52i)T - 2iT^{2} \)
3 \( 1 + (1.06 + 2.56i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (2.90 + 1.20i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (4.70 + 1.94i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.39T + 13T^{2} \)
19 \( 1 + (-3.63 - 3.63i)T + 19iT^{2} \)
23 \( 1 + (2.37 - 5.73i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (2.65 + 6.41i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-2.33 + 0.966i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-2.97 - 7.17i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.83 - 9.25i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (2.08 + 2.08i)T + 43iT^{2} \)
47 \( 1 - 5.08T + 47T^{2} \)
53 \( 1 + (2.93 - 2.93i)T - 53iT^{2} \)
59 \( 1 + (-0.594 + 0.594i)T - 59iT^{2} \)
61 \( 1 + (2.29 - 5.54i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 4.10iT - 67T^{2} \)
71 \( 1 + (9.66 - 4.00i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-0.549 + 0.227i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (6.77 + 2.80i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (3.35 - 3.35i)T - 83iT^{2} \)
89 \( 1 - 14.1iT - 89T^{2} \)
97 \( 1 + (-2.37 + 0.982i)T + (68.5 - 68.5i)T^{2} \)
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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

−11.50472324563020133454219322416, −10.23541767295141294448232626542, −9.701609641826944590776551376437, −8.110617165643095080608253341945, −7.85986409663544261330567908948, −7.00855601327474220909013407317, −5.97275331860395561108047625405, −5.73462917992855241908139912850, −3.22360269701160123005405295751, −1.18068411114736070104800835568, 0.15931479322803558033775783271, 2.64705570336891495145182619598, 3.49312697123903755317043223819, 4.97702338311747777738277336377, 5.82292520879262857182781697754, 7.42407121760542920297032496942, 8.767577065406845666615150019741, 9.398863745805661398044020829058, 10.17217820024102521428100719683, 10.48497940357391262011604209525

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