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Properties

Label 2-425-17.4-c1-0-17
Degree $2$
Conductor $425$
Sign $-0.615 + 0.788i$
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  − i·2-s + (−1 + i)3-s + 4-s + (1 + i)6-s + (−3 − 3i)7-s − 3i·8-s + i·9-s + (−3 − 3i)11-s + (−1 + i)12-s + (−3 + 3i)14-s − 16-s + (4 − i)17-s + 18-s − 6i·19-s + 6·21-s + (−3 + 3i)22-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.577 + 0.577i)3-s + 0.5·4-s + (0.408 + 0.408i)6-s + (−1.13 − 1.13i)7-s − 1.06i·8-s + 0.333i·9-s + (−0.904 − 0.904i)11-s + (−0.288 + 0.288i)12-s + (−0.801 + 0.801i)14-s − 0.250·16-s + (0.970 − 0.242i)17-s + 0.235·18-s − 1.37i·19-s + 1.30·21-s + (−0.639 + 0.639i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (276, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.615 + 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.398624 - 0.816972i\)
\(L(\frac12)\) \(\approx\) \(0.398624 - 0.816972i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4 + i)T \)
good2 \( 1 + iT - 2T^{2} \)
3 \( 1 + (1 - i)T - 3iT^{2} \)
7 \( 1 + (3 + 3i)T + 7iT^{2} \)
11 \( 1 + (3 + 3i)T + 11iT^{2} \)
13 \( 1 + 13T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 + (1 - i)T - 31iT^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + (3 + 3i)T + 41iT^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + (-1 - i)T + 61iT^{2} \)
67 \( 1 - 6T + 67T^{2} \)
71 \( 1 + (-3 + 3i)T - 71iT^{2} \)
73 \( 1 + (-3 + 3i)T - 73iT^{2} \)
79 \( 1 + (-7 - 7i)T + 79iT^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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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

−10.85288534787206863613664780116, −10.17638680979767677528936888510, −9.637382338936088951922805099041, −8.023010886338626353670450989520, −7.01810061752613472156754052628, −6.11787150208091906409812956620, −4.90685685004734445533013436417, −3.62961153618714600773447475715, −2.74580015880356677509674999510, −0.58750809406244438518557466809, 2.02401108121806693620180645918, 3.34266943572032839851220758708, 5.43728157632445237146996716750, 5.84959385588803731290290142881, 6.76714988496951488910683409710, 7.52990186095398680341139091394, 8.554667927277463650858942720792, 9.735353948737950290286309512124, 10.52651583056021977505408685280, 11.92351263353758655517547923465

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