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Properties

Label 2-425-85.19-c1-0-7
Degree $2$
Conductor $425$
Sign $-0.637 - 0.770i$
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  + (−1.66 + 1.66i)2-s + (0.600 + 1.44i)3-s − 3.53i·4-s + (−3.41 − 1.41i)6-s + (3.30 + 1.37i)7-s + (2.54 + 2.54i)8-s + (0.379 − 0.379i)9-s + (2.29 + 0.950i)11-s + (5.12 − 2.12i)12-s + 1.25·13-s + (−7.78 + 3.22i)14-s − 1.40·16-s + (1.48 − 3.84i)17-s + 1.26i·18-s + (2.56 + 2.56i)19-s + ⋯
L(s)  = 1  + (−1.17 + 1.17i)2-s + (0.346 + 0.837i)3-s − 1.76i·4-s + (−1.39 − 0.576i)6-s + (1.25 + 0.518i)7-s + (0.900 + 0.900i)8-s + (0.126 − 0.126i)9-s + (0.691 + 0.286i)11-s + (1.47 − 0.612i)12-s + 0.349·13-s + (−2.08 + 0.861i)14-s − 0.352·16-s + (0.360 − 0.932i)17-s + 0.297i·18-s + (0.589 + 0.589i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.637 - 0.770i$
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: \((2,\ 425,\ (\ :1/2),\ -0.637 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.441209 + 0.938001i\)
\(L(\frac12)\) \(\approx\) \(0.441209 + 0.938001i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 3.84i)T \)
good2 \( 1 + (1.66 - 1.66i)T - 2iT^{2} \)
3 \( 1 + (-0.600 - 1.44i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (-3.30 - 1.37i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-2.29 - 0.950i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.25T + 13T^{2} \)
19 \( 1 + (-2.56 - 2.56i)T + 19iT^{2} \)
23 \( 1 + (3.38 - 8.16i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (3.35 + 8.09i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-2.25 + 0.935i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (1.76 + 4.25i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.65 - 3.99i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (5.33 + 5.33i)T + 43iT^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + (-4.02 + 4.02i)T - 53iT^{2} \)
59 \( 1 + (3.16 - 3.16i)T - 59iT^{2} \)
61 \( 1 + (-0.0929 + 0.224i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 7.23iT - 67T^{2} \)
71 \( 1 + (1.69 - 0.703i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (5.05 - 2.09i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-8.34 - 3.45i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (5.17 - 5.17i)T - 83iT^{2} \)
89 \( 1 - 2.19iT - 89T^{2} \)
97 \( 1 + (-9.07 + 3.75i)T + (68.5 - 68.5i)T^{2} \)
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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

−11.35772640993556340274598720216, −9.891507138929912623505383501345, −9.680330906661907119777561878744, −8.723473117662685832882766442955, −7.970879003766134251272685788736, −7.18072066098933694449730091565, −5.91307984682464701119957823699, −5.03618059646237345819776431720, −3.72093389348558841984428772072, −1.53376318802040265692202563628, 1.18487065680015306737331041911, 1.88584523006573766673731459019, 3.34073138197306968055319683547, 4.70928135031962541908904857361, 6.54096067588431227104592406668, 7.64395174551498752230529619525, 8.290902486929160580536920618983, 8.871714434246994702568851406962, 10.19113486591191993502616134393, 10.77015528606560385727378166195

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