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Properties

Label 2-425-17.13-c1-0-14
Degree $2$
Conductor $425$
Sign $-0.0534 + 0.998i$
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  − 2.08i·2-s + (1.31 + 1.31i)3-s − 2.34·4-s + (2.73 − 2.73i)6-s + (0.971 − 0.971i)7-s + 0.713i·8-s + 0.435i·9-s + (−0.384 + 0.384i)11-s + (−3.07 − 3.07i)12-s + 5.39·13-s + (−2.02 − 2.02i)14-s − 3.19·16-s + (2.36 + 3.38i)17-s + 0.907·18-s − 4.86i·19-s + ⋯
L(s)  = 1  − 1.47i·2-s + (0.756 + 0.756i)3-s − 1.17·4-s + (1.11 − 1.11i)6-s + (0.367 − 0.367i)7-s + 0.252i·8-s + 0.145i·9-s + (−0.116 + 0.116i)11-s + (−0.886 − 0.886i)12-s + 1.49·13-s + (−0.541 − 0.541i)14-s − 0.799·16-s + (0.572 + 0.819i)17-s + 0.213·18-s − 1.11i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0534 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0534 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.0534 + 0.998i$
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: \((2,\ 425,\ (\ :1/2),\ -0.0534 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25877 - 1.32795i\)
\(L(\frac12)\) \(\approx\) \(1.25877 - 1.32795i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.36 - 3.38i)T \)
good2 \( 1 + 2.08iT - 2T^{2} \)
3 \( 1 + (-1.31 - 1.31i)T + 3iT^{2} \)
7 \( 1 + (-0.971 + 0.971i)T - 7iT^{2} \)
11 \( 1 + (0.384 - 0.384i)T - 11iT^{2} \)
13 \( 1 - 5.39T + 13T^{2} \)
19 \( 1 + 4.86iT - 19T^{2} \)
23 \( 1 + (1.26 - 1.26i)T - 23iT^{2} \)
29 \( 1 + (-1.29 - 1.29i)T + 29iT^{2} \)
31 \( 1 + (5.73 + 5.73i)T + 31iT^{2} \)
37 \( 1 + (4.22 + 4.22i)T + 37iT^{2} \)
41 \( 1 + (2.70 - 2.70i)T - 41iT^{2} \)
43 \( 1 - 3.66iT - 43T^{2} \)
47 \( 1 - 9.07T + 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 - 6.52iT - 59T^{2} \)
61 \( 1 + (10.9 - 10.9i)T - 61iT^{2} \)
67 \( 1 + 5.68T + 67T^{2} \)
71 \( 1 + (-0.749 - 0.749i)T + 71iT^{2} \)
73 \( 1 + (10.3 + 10.3i)T + 73iT^{2} \)
79 \( 1 + (0.878 - 0.878i)T - 79iT^{2} \)
83 \( 1 - 13.5iT - 83T^{2} \)
89 \( 1 + 0.989T + 89T^{2} \)
97 \( 1 + (8.05 + 8.05i)T + 97iT^{2} \)
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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

−10.75136064196120114188115288970, −10.34904772791800502710791980765, −9.189060151154244291075400732752, −8.805736256233870287102505428270, −7.52710016257943327203316753492, −6.01162831805840671925921710958, −4.39233665642041340413505496754, −3.77383242639562350000789181478, −2.81962901234300191851026446592, −1.33355272478612962401680801780, 1.81281971701755226449434383201, 3.43778967212871711027567830674, 5.06064141975667493243926560242, 5.92305136903408444283396091701, 6.90760516253253399157257802697, 7.72461100060564803808713142199, 8.447789447184201059949885903023, 8.893481162579636768617136319396, 10.37620022183034305648885246708, 11.52103087547918781841369091087

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