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Properties

Label 2-325-13.5-c2-0-17
Degree $2$
Conductor $325$
Sign $0.996 - 0.0871i$
Analytic cond. $8.85560$
Root an. cond. $2.97583$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.67 − 1.67i)2-s − 0.982·3-s − 1.59i·4-s + (−1.64 + 1.64i)6-s + (6.39 + 6.39i)7-s + (4.01 + 4.01i)8-s − 8.03·9-s + (11.2 + 11.2i)11-s + 1.57i·12-s + (−12.7 − 2.66i)13-s + 21.4·14-s + 19.8·16-s + 22.8i·17-s + (−13.4 + 13.4i)18-s + (13.7 − 13.7i)19-s + ⋯
L(s)  = 1  + (0.836 − 0.836i)2-s − 0.327·3-s − 0.399i·4-s + (−0.273 + 0.273i)6-s + (0.913 + 0.913i)7-s + (0.502 + 0.502i)8-s − 0.892·9-s + (1.02 + 1.02i)11-s + 0.130i·12-s + (−0.978 − 0.205i)13-s + 1.52·14-s + 1.23·16-s + 1.34i·17-s + (−0.746 + 0.746i)18-s + (0.721 − 0.721i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.996 - 0.0871i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ 0.996 - 0.0871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.43861 + 0.106444i\)
\(L(\frac12)\) \(\approx\) \(2.43861 + 0.106444i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + (12.7 + 2.66i)T \)
good2 \( 1 + (-1.67 + 1.67i)T - 4iT^{2} \)
3 \( 1 + 0.982T + 9T^{2} \)
7 \( 1 + (-6.39 - 6.39i)T + 49iT^{2} \)
11 \( 1 + (-11.2 - 11.2i)T + 121iT^{2} \)
17 \( 1 - 22.8iT - 289T^{2} \)
19 \( 1 + (-13.7 + 13.7i)T - 361iT^{2} \)
23 \( 1 + 6.04iT - 529T^{2} \)
29 \( 1 - 7.49T + 841T^{2} \)
31 \( 1 + (-30.7 + 30.7i)T - 961iT^{2} \)
37 \( 1 + (18.2 + 18.2i)T + 1.36e3iT^{2} \)
41 \( 1 + (-18.0 + 18.0i)T - 1.68e3iT^{2} \)
43 \( 1 - 75.9iT - 1.84e3T^{2} \)
47 \( 1 + (24.9 + 24.9i)T + 2.20e3iT^{2} \)
53 \( 1 + 55.7T + 2.80e3T^{2} \)
59 \( 1 + (-26.2 - 26.2i)T + 3.48e3iT^{2} \)
61 \( 1 + 94.1T + 3.72e3T^{2} \)
67 \( 1 + (1.31 - 1.31i)T - 4.48e3iT^{2} \)
71 \( 1 + (-44.7 + 44.7i)T - 5.04e3iT^{2} \)
73 \( 1 + (-34.3 - 34.3i)T + 5.32e3iT^{2} \)
79 \( 1 + 115.T + 6.24e3T^{2} \)
83 \( 1 + (-99.1 + 99.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (26.6 + 26.6i)T + 7.92e3iT^{2} \)
97 \( 1 + (35.7 - 35.7i)T - 9.40e3iT^{2} \)
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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

−11.67981382571029259855314071578, −10.92467914470782862281209671410, −9.734877559641610788519069178354, −8.601624251663177217826452179850, −7.66744709142546263290800523165, −6.17050225060291512341208836537, −5.09623544506775204407368690342, −4.37661277654615935799173920380, −2.85027354149063982114820745593, −1.79944552180775155468987824455, 1.02216605012289798426793329287, 3.34062594351943367926727614595, 4.64452871905580572327759752562, 5.33022802751946107968005133726, 6.42127727422034679960894449947, 7.27738064700644096437890679554, 8.219979195858287966371987465374, 9.507151656912324723222457056535, 10.62699106398276954845466846098, 11.55968295211980099390780463576

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