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Properties

Label 2-425-85.49-c1-0-14
Degree $2$
Conductor $425$
Sign $0.550 + 0.835i$
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  + (1.44 − 1.44i)2-s + (−2.95 + 1.22i)3-s − 2.18i·4-s + (−2.50 + 6.03i)6-s + (−0.450 + 1.08i)7-s + (−0.263 − 0.263i)8-s + (5.09 − 5.09i)9-s + (1.88 − 4.54i)11-s + (2.66 + 6.44i)12-s + 2.46·13-s + (0.921 + 2.22i)14-s + 3.60·16-s + (3.89 − 1.36i)17-s − 14.7i·18-s + (1.44 + 1.44i)19-s + ⋯
L(s)  = 1  + (1.02 − 1.02i)2-s + (−1.70 + 0.706i)3-s − 1.09i·4-s + (−1.02 + 2.46i)6-s + (−0.170 + 0.411i)7-s + (−0.0931 − 0.0931i)8-s + (1.69 − 1.69i)9-s + (0.567 − 1.37i)11-s + (0.770 + 1.85i)12-s + 0.683·13-s + (0.246 + 0.594i)14-s + 0.900·16-s + (0.943 − 0.330i)17-s − 3.47i·18-s + (0.331 + 0.331i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.550 + 0.835i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.550 + 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31842 - 0.710169i\)
\(L(\frac12)\) \(\approx\) \(1.31842 - 0.710169i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.89 + 1.36i)T \)
good2 \( 1 + (-1.44 + 1.44i)T - 2iT^{2} \)
3 \( 1 + (2.95 - 1.22i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (0.450 - 1.08i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.88 + 4.54i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 2.46T + 13T^{2} \)
19 \( 1 + (-1.44 - 1.44i)T + 19iT^{2} \)
23 \( 1 + (-0.109 - 0.0455i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.984 - 0.407i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.06 - 2.58i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (2.13 - 0.885i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.662 + 0.274i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (7.13 + 7.13i)T + 43iT^{2} \)
47 \( 1 + 3.39T + 47T^{2} \)
53 \( 1 + (-9.84 + 9.84i)T - 53iT^{2} \)
59 \( 1 + (1.07 - 1.07i)T - 59iT^{2} \)
61 \( 1 + (-7.46 - 3.09i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 4.92iT - 67T^{2} \)
71 \( 1 + (-2.53 - 6.13i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (1.40 + 3.38i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (3.13 - 7.55i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (4.30 - 4.30i)T - 83iT^{2} \)
89 \( 1 + 8.46iT - 89T^{2} \)
97 \( 1 + (1.71 + 4.14i)T + (-68.5 + 68.5i)T^{2} \)
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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

−11.35587195925883189046273403977, −10.52505165400800734076738022587, −9.838455319350271716709430100425, −8.535625009128286265736648786509, −6.73579563830629869273236875908, −5.62229532557319033352180788092, −5.39837444136033258716398439216, −4.03339764463689725063026586581, −3.31741739835759339850609216982, −1.11038250645251968547076724887, 1.33997222200763738301375314793, 3.96275842548863895628544637880, 4.88442510698282404124103815262, 5.70286878034356913423375331296, 6.54258680109066430863922825060, 7.09042517240848524651956965884, 7.88057625132665371967559314240, 9.799279351911628443269925628602, 10.59221242787113999481567504148, 11.71345434167423060412125896983

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