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Properties

Label 2-425-17.13-c1-0-1
Degree $2$
Conductor $425$
Sign $-0.957 - 0.289i$
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  + 2.07i·2-s + (−1.78 − 1.78i)3-s − 2.28·4-s + (3.69 − 3.69i)6-s + (0.260 − 0.260i)7-s − 0.595i·8-s + 3.36i·9-s + (−1.76 + 1.76i)11-s + (4.08 + 4.08i)12-s + 4.68·13-s + (0.540 + 0.540i)14-s − 3.34·16-s + (−3.84 + 1.48i)17-s − 6.97·18-s + 7.16i·19-s + ⋯
L(s)  = 1  + 1.46i·2-s + (−1.03 − 1.03i)3-s − 1.14·4-s + (1.50 − 1.50i)6-s + (0.0986 − 0.0986i)7-s − 0.210i·8-s + 1.12i·9-s + (−0.532 + 0.532i)11-s + (1.17 + 1.17i)12-s + 1.30·13-s + (0.144 + 0.144i)14-s − 0.835·16-s + (−0.932 + 0.360i)17-s − 1.64·18-s + 1.64i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.957 - 0.289i$
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: \((2,\ 425,\ (\ :1/2),\ -0.957 - 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0927958 + 0.626893i\)
\(L(\frac12)\) \(\approx\) \(0.0927958 + 0.626893i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.84 - 1.48i)T \)
good2 \( 1 - 2.07iT - 2T^{2} \)
3 \( 1 + (1.78 + 1.78i)T + 3iT^{2} \)
7 \( 1 + (-0.260 + 0.260i)T - 7iT^{2} \)
11 \( 1 + (1.76 - 1.76i)T - 11iT^{2} \)
13 \( 1 - 4.68T + 13T^{2} \)
19 \( 1 - 7.16iT - 19T^{2} \)
23 \( 1 + (4.73 - 4.73i)T - 23iT^{2} \)
29 \( 1 + (4.79 + 4.79i)T + 29iT^{2} \)
31 \( 1 + (-3.40 - 3.40i)T + 31iT^{2} \)
37 \( 1 + (-1.37 - 1.37i)T + 37iT^{2} \)
41 \( 1 + (-1.66 + 1.66i)T - 41iT^{2} \)
43 \( 1 - 11.7iT - 43T^{2} \)
47 \( 1 - 1.65T + 47T^{2} \)
53 \( 1 + 6.81iT - 53T^{2} \)
59 \( 1 + 0.484iT - 59T^{2} \)
61 \( 1 + (-4.86 + 4.86i)T - 61iT^{2} \)
67 \( 1 - 1.87T + 67T^{2} \)
71 \( 1 + (-1.21 - 1.21i)T + 71iT^{2} \)
73 \( 1 + (0.202 + 0.202i)T + 73iT^{2} \)
79 \( 1 + (-3.80 + 3.80i)T - 79iT^{2} \)
83 \( 1 - 9.94iT - 83T^{2} \)
89 \( 1 + 4.30T + 89T^{2} \)
97 \( 1 + (9.01 + 9.01i)T + 97iT^{2} \)
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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

−11.59809718327628569063631343246, −10.90858340931274775886810446985, −9.624211600062659587103656690398, −8.184222677648773413021326338861, −7.81689570937550426145412206666, −6.73366644460602954234824735817, −6.07927571486740227209958893553, −5.50578810535654991705719483309, −4.14373628099135465223505056796, −1.72211865247491794086935508072, 0.45878954507378967587357711680, 2.44791485572697754172012076659, 3.79095893092929381277233356797, 4.58804464149939646880023162116, 5.64799129982182478469890799186, 6.76157478397282389371695667056, 8.582855394948275212361298319782, 9.303038830987056504183484595935, 10.34590135733983997542488987920, 10.93719109052559768809169540282

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