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Properties

Label 2-425-85.74-c2-0-3
Degree $2$
Conductor $425$
Sign $-0.351 + 0.936i$
Analytic cond. $11.5804$
Root an. cond. $3.40300$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.675 + 1.63i)2-s + (0.789 + 3.96i)3-s + (0.624 + 0.624i)4-s + (−7.00 − 1.39i)6-s + (−3.40 + 2.27i)7-s + (−7.96 + 3.29i)8-s + (−6.81 + 2.82i)9-s + (1.35 − 6.80i)11-s + (−1.98 + 2.97i)12-s + (2.37 + 2.37i)13-s + (−1.40 − 7.08i)14-s − 11.6i·16-s + (−14.5 + 8.76i)17-s − 13.0i·18-s + (−22.7 − 9.43i)19-s + ⋯
L(s)  = 1  + (−0.337 + 0.815i)2-s + (0.263 + 1.32i)3-s + (0.156 + 0.156i)4-s + (−1.16 − 0.232i)6-s + (−0.486 + 0.324i)7-s + (−0.995 + 0.412i)8-s + (−0.757 + 0.313i)9-s + (0.123 − 0.618i)11-s + (−0.165 + 0.247i)12-s + (0.182 + 0.182i)13-s + (−0.100 − 0.506i)14-s − 0.730i·16-s + (−0.856 + 0.515i)17-s − 0.723i·18-s + (−1.19 − 0.496i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.351 + 0.936i$
Analytic conductor: \(11.5804\)
Root analytic conductor: \(3.40300\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1),\ -0.351 + 0.936i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.534935 - 0.771929i\)
\(L(\frac12)\) \(\approx\) \(0.534935 - 0.771929i\)
\(L(2)\) 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 \( 1 + (14.5 - 8.76i)T \)
good2 \( 1 + (0.675 - 1.63i)T + (-2.82 - 2.82i)T^{2} \)
3 \( 1 + (-0.789 - 3.96i)T + (-8.31 + 3.44i)T^{2} \)
7 \( 1 + (3.40 - 2.27i)T + (18.7 - 45.2i)T^{2} \)
11 \( 1 + (-1.35 + 6.80i)T + (-111. - 46.3i)T^{2} \)
13 \( 1 + (-2.37 - 2.37i)T + 169iT^{2} \)
19 \( 1 + (22.7 + 9.43i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-2.24 + 11.2i)T + (-488. - 202. i)T^{2} \)
29 \( 1 + (9.98 + 6.67i)T + (321. + 776. i)T^{2} \)
31 \( 1 + (-7.38 - 37.1i)T + (-887. + 367. i)T^{2} \)
37 \( 1 + (-6.29 - 31.6i)T + (-1.26e3 + 523. i)T^{2} \)
41 \( 1 + (-18.7 - 28.1i)T + (-643. + 1.55e3i)T^{2} \)
43 \( 1 + (-1.33 - 3.21i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + (3.16 + 3.16i)T + 2.20e3iT^{2} \)
53 \( 1 + (-11.7 + 28.3i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-4.49 - 10.8i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-58.9 + 39.3i)T + (1.42e3 - 3.43e3i)T^{2} \)
67 \( 1 - 28.5T + 4.48e3T^{2} \)
71 \( 1 + (33.7 - 6.70i)T + (4.65e3 - 1.92e3i)T^{2} \)
73 \( 1 + (77.8 + 52.0i)T + (2.03e3 + 4.92e3i)T^{2} \)
79 \( 1 + (4.18 - 21.0i)T + (-5.76e3 - 2.38e3i)T^{2} \)
83 \( 1 + (-104. - 43.4i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-28.3 - 28.3i)T + 7.92e3iT^{2} \)
97 \( 1 + (92.4 - 138. i)T + (-3.60e3 - 8.69e3i)T^{2} \)
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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

−11.27825819284177175814599108997, −10.61741442288718382928101560256, −9.513139053122266904488449740295, −8.784361749852645156392442568406, −8.276073908858038420482546937079, −6.76280327609385136646600943387, −6.14913354059508477743354222947, −4.84269278319889831637588048469, −3.72523083302433975606321376038, −2.65474811953527633567374649444, 0.40303329379820524729047492426, 1.76702258854957230669175053276, 2.57129609833347887981869702409, 4.02801910890211054377869966207, 5.87825481142189564427563932304, 6.73377457270257867947847156356, 7.42406466785865069795821705781, 8.607203092524353539626927133196, 9.526575456425139431677236671986, 10.41623802889141235166290094353

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