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Properties

Label 2-325-13.10-c1-0-11
Degree $2$
Conductor $325$
Sign $0.964 - 0.265i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
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  + (1.5 − 0.866i)2-s + (1 + 1.73i)3-s + (0.5 − 0.866i)4-s + (3 + 1.73i)6-s + 1.73i·8-s + (−0.499 + 0.866i)9-s + 2·12-s + (2.5 − 2.59i)13-s + (2.49 + 4.33i)16-s + (−1.5 + 2.59i)17-s + 1.73i·18-s + (−3 − 1.73i)19-s + (−3 − 5.19i)23-s + (−2.99 + 1.73i)24-s + (1.5 − 6.06i)26-s + 4.00·27-s + ⋯
L(s)  = 1  + (1.06 − 0.612i)2-s + (0.577 + 0.999i)3-s + (0.250 − 0.433i)4-s + (1.22 + 0.707i)6-s + 0.612i·8-s + (−0.166 + 0.288i)9-s + 0.577·12-s + (0.693 − 0.720i)13-s + (0.624 + 1.08i)16-s + (−0.363 + 0.630i)17-s + 0.408i·18-s + (−0.688 − 0.397i)19-s + (−0.625 − 1.08i)23-s + (−0.612 + 0.353i)24-s + (0.294 − 1.18i)26-s + 0.769·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53310 + 0.341754i\)
\(L(\frac12)\) \(\approx\) \(2.53310 + 0.341754i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + (-2.5 + 2.59i)T \)
good2 \( 1 + (-1.5 + 0.866i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (7.5 - 4.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-6 - 3.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 1.73i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6 + 3.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.71670396963559990504457926355, −10.68006628426900369941810750173, −10.15538143203771762204339424550, −8.765344190845017559391492768609, −8.260113470217641071535381697909, −6.47138951629164004319458304994, −5.27685297911296615657841987840, −4.19843180921805728322683579281, −3.59011244658650847893802284355, −2.36543313134968192161402685996, 1.71764745575139670633268757715, 3.36148963433361345498900085481, 4.53821836070801265433574188228, 5.77398399297014933904924571342, 6.73250033221242312623278317749, 7.39633874029440911504684500765, 8.487785112214251969283663091759, 9.512541038628644593229302144859, 10.82646376404256733503150508582, 12.03683157845317867412048565630

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