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Properties

Label 2-325-65.34-c2-0-1
Degree $2$
Conductor $325$
Sign $-0.282 + 0.959i$
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  + (−0.581 + 0.581i)2-s + 4.16i·3-s + 3.32i·4-s + (−2.41 − 2.41i)6-s + (−4.58 − 4.58i)7-s + (−4.25 − 4.25i)8-s − 8.32·9-s + (5.32 − 5.32i)11-s − 13.8·12-s + (−11.5 + 5.90i)13-s + 5.32·14-s − 8.35·16-s − 21.9·17-s + (4.83 − 4.83i)18-s + (−3.16 − 3.16i)19-s + ⋯
L(s)  = 1  + (−0.290 + 0.290i)2-s + 1.38i·3-s + 0.831i·4-s + (−0.403 − 0.403i)6-s + (−0.654 − 0.654i)7-s + (−0.532 − 0.532i)8-s − 0.924·9-s + (0.484 − 0.484i)11-s − 1.15·12-s + (−0.890 + 0.454i)13-s + 0.380·14-s − 0.521·16-s − 1.29·17-s + (0.268 − 0.268i)18-s + (−0.166 − 0.166i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.282 + 0.959i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ -0.282 + 0.959i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.211229 - 0.282310i\)
\(L(\frac12)\) \(\approx\) \(0.211229 - 0.282310i\)
\(L(2)\) 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 \( 1 + (11.5 - 5.90i)T \)
good2 \( 1 + (0.581 - 0.581i)T - 4iT^{2} \)
3 \( 1 - 4.16iT - 9T^{2} \)
7 \( 1 + (4.58 + 4.58i)T + 49iT^{2} \)
11 \( 1 + (-5.32 + 5.32i)T - 121iT^{2} \)
17 \( 1 + 21.9T + 289T^{2} \)
19 \( 1 + (3.16 + 3.16i)T + 361iT^{2} \)
23 \( 1 - 8.51T + 529T^{2} \)
29 \( 1 - 5.81T + 841T^{2} \)
31 \( 1 + (-0.513 - 0.513i)T + 961iT^{2} \)
37 \( 1 + (-24.2 - 24.2i)T + 1.36e3iT^{2} \)
41 \( 1 + (-4.83 - 4.83i)T + 1.68e3iT^{2} \)
43 \( 1 + 30.4T + 1.84e3T^{2} \)
47 \( 1 + (37.3 + 37.3i)T + 2.20e3iT^{2} \)
53 \( 1 - 35.8iT - 2.80e3T^{2} \)
59 \( 1 + (58.2 - 58.2i)T - 3.48e3iT^{2} \)
61 \( 1 + 80.3T + 3.72e3T^{2} \)
67 \( 1 + (39.0 - 39.0i)T - 4.48e3iT^{2} \)
71 \( 1 + (-91.5 - 91.5i)T + 5.04e3iT^{2} \)
73 \( 1 + (31.6 + 31.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 18.7T + 6.24e3T^{2} \)
83 \( 1 + (44.6 - 44.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (8.89 - 8.89i)T - 7.92e3iT^{2} \)
97 \( 1 + (-121. + 121. i)T - 9.40e3iT^{2} \)
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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

−11.83364778842964156472811428476, −10.99271839831948237263248600104, −9.982595478806816196998112830045, −9.273601946607716115711307735365, −8.565185055738632383832807169898, −7.21973122053792851598551223527, −6.41852856622058682380676384577, −4.70597776011152355217333460587, −3.97880264031952622157885697446, −2.93857183960134478380748178092, 0.17342376913020899698506241571, 1.73732129860011342802034846804, 2.67437552092326385019373380760, 4.83239035832288416477484086890, 6.15361862833748922876692651148, 6.67348485159856578083006874285, 7.79776459834229318604516888283, 9.008254475570911381406536010890, 9.665784401345163227659023355467, 10.81467648401389125314747525333

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