Properties

Label 2-425-17.16-c3-0-71
Degree $2$
Conductor $425$
Sign $-0.174 + 0.984i$
Analytic cond. $25.0758$
Root an. cond. $5.00757$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.37·2-s + 7.36i·3-s + 3.37·4-s + 24.8i·6-s − 17.4i·7-s − 15.6·8-s − 27.2·9-s − 51.5i·11-s + 24.8i·12-s − 75.2·13-s − 58.9i·14-s − 79.6·16-s + (12.2 − 69.0i)17-s − 91.8·18-s − 28·19-s + ⋯
L(s)  = 1  + 1.19·2-s + 1.41i·3-s + 0.421·4-s + 1.68i·6-s − 0.943i·7-s − 0.689·8-s − 1.00·9-s − 1.41i·11-s + 0.597i·12-s − 1.60·13-s − 1.12i·14-s − 1.24·16-s + (0.174 − 0.984i)17-s − 1.20·18-s − 0.338·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.174 + 0.984i$
Analytic conductor: \(25.0758\)
Root analytic conductor: \(5.00757\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :3/2),\ -0.174 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9663659910\)
\(L(\frac12)\) \(\approx\) \(0.9663659910\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.2 + 69.0i)T \)
good2 \( 1 - 3.37T + 8T^{2} \)
3 \( 1 - 7.36iT - 27T^{2} \)
7 \( 1 + 17.4iT - 343T^{2} \)
11 \( 1 + 51.5iT - 1.33e3T^{2} \)
13 \( 1 + 75.2T + 2.19e3T^{2} \)
19 \( 1 + 28T + 6.85e3T^{2} \)
23 \( 1 - 19.1iT - 1.21e4T^{2} \)
29 \( 1 - 70.7iT - 2.43e4T^{2} \)
31 \( 1 - 41.4iT - 2.97e4T^{2} \)
37 \( 1 + 135. iT - 5.06e4T^{2} \)
41 \( 1 - 288. iT - 6.89e4T^{2} \)
43 \( 1 + 88.2T + 7.95e4T^{2} \)
47 \( 1 + 157.T + 1.03e5T^{2} \)
53 \( 1 + 120.T + 1.48e5T^{2} \)
59 \( 1 + 696.T + 2.05e5T^{2} \)
61 \( 1 + 683. iT - 2.26e5T^{2} \)
67 \( 1 + 123.T + 3.00e5T^{2} \)
71 \( 1 - 225. iT - 3.57e5T^{2} \)
73 \( 1 + 919. iT - 3.89e5T^{2} \)
79 \( 1 + 354. iT - 4.93e5T^{2} \)
83 \( 1 - 955.T + 5.71e5T^{2} \)
89 \( 1 - 617.T + 7.04e5T^{2} \)
97 \( 1 - 428. iT - 9.12e5T^{2} \)
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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

−10.62290353977539911452079953624, −9.683555526835593196192335364647, −9.005799860583317784123243562422, −7.62958694075698906486270221891, −6.37448369584037951678784924890, −5.14742351689751084623101779631, −4.71145478897404407931429842764, −3.66254797152423182490380305926, −2.94350644163360497697122408866, −0.19478885833732282037913097519, 1.94531286721195205959549242991, 2.64260720346395083423608555018, 4.31886824722300914881225482185, 5.28460821178331285771223616379, 6.24356918822843067233695223501, 7.06126621242062110622866513951, 7.961713770189312430901541706438, 9.106366266510760216124085963136, 10.12397857796328210761648470130, 11.74596182181200939029611799815

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