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Properties

Label 2-425-85.4-c1-0-10
Degree $2$
Conductor $425$
Sign $0.722 + 0.691i$
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  + 0.0601·2-s + (0.294 + 0.294i)3-s − 1.99·4-s + (0.0177 + 0.0177i)6-s + (0.900 − 0.900i)7-s − 0.240·8-s − 2.82i·9-s + (2.24 + 2.24i)11-s + (−0.588 − 0.588i)12-s − 4.23i·13-s + (0.0542 − 0.0542i)14-s + 3.97·16-s + (1.29 − 3.91i)17-s − 0.170i·18-s + 4.76i·19-s + ⋯
L(s)  = 1  + 0.0425·2-s + (0.170 + 0.170i)3-s − 0.998·4-s + (0.00724 + 0.00724i)6-s + (0.340 − 0.340i)7-s − 0.0850·8-s − 0.942i·9-s + (0.677 + 0.677i)11-s + (−0.169 − 0.169i)12-s − 1.17i·13-s + (0.0144 − 0.0144i)14-s + 0.994·16-s + (0.313 − 0.949i)17-s − 0.0400i·18-s + 1.09i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14693 - 0.460350i\)
\(L(\frac12)\) \(\approx\) \(1.14693 - 0.460350i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 + 3.91i)T \)
good2 \( 1 - 0.0601T + 2T^{2} \)
3 \( 1 + (-0.294 - 0.294i)T + 3iT^{2} \)
7 \( 1 + (-0.900 + 0.900i)T - 7iT^{2} \)
11 \( 1 + (-2.24 - 2.24i)T + 11iT^{2} \)
13 \( 1 + 4.23iT - 13T^{2} \)
19 \( 1 - 4.76iT - 19T^{2} \)
23 \( 1 + (-5.13 + 5.13i)T - 23iT^{2} \)
29 \( 1 + (-1.35 + 1.35i)T - 29iT^{2} \)
31 \( 1 + (-1.64 + 1.64i)T - 31iT^{2} \)
37 \( 1 + (3.84 + 3.84i)T + 37iT^{2} \)
41 \( 1 + (-0.0814 - 0.0814i)T + 41iT^{2} \)
43 \( 1 - 0.562T + 43T^{2} \)
47 \( 1 - 5.01iT - 47T^{2} \)
53 \( 1 + 7.75T + 53T^{2} \)
59 \( 1 + 2.01iT - 59T^{2} \)
61 \( 1 + (7.03 + 7.03i)T + 61iT^{2} \)
67 \( 1 + 3.64iT - 67T^{2} \)
71 \( 1 + (6.92 - 6.92i)T - 71iT^{2} \)
73 \( 1 + (-10.2 - 10.2i)T + 73iT^{2} \)
79 \( 1 + (-5.78 - 5.78i)T + 79iT^{2} \)
83 \( 1 + 4.38T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + (-9.15 - 9.15i)T + 97iT^{2} \)
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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

−10.94038964287919573536877749968, −9.930398537576771604894742722873, −9.375270521417338982483153872785, −8.420441294317865607220006783108, −7.52116676789782005033380647640, −6.28584455825804951911474014954, −5.09394308542268934787887749218, −4.17785772344443281676989108454, −3.14949022201259418413237136333, −0.912986964978248285360091220641, 1.56396774354657618709570155889, 3.30531576959406078227662368440, 4.54624634832076738600416983373, 5.34344761728791563051645479484, 6.60238956414001196649566319923, 7.78314379440063337471913588650, 8.770563783773156590948910227484, 9.135902105801427738194914383454, 10.39802954860058143657687095508, 11.32006001008337060304229130112

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