Properties

Label 2-325-13.10-c1-0-18
Degree $2$
Conductor $325$
Sign $-0.804 + 0.593i$
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  + (2.16 − 1.24i)2-s + (−1.41 − 2.44i)3-s + (2.11 − 3.66i)4-s + (−6.10 − 3.52i)6-s + (1.64 + 0.952i)7-s − 5.55i·8-s + (−2.49 + 4.32i)9-s + (0.926 − 0.534i)11-s − 11.9·12-s + (−1.40 + 3.32i)13-s + 4.75·14-s + (−2.70 − 4.69i)16-s + (−0.318 + 0.551i)17-s + 12.4i·18-s + (4.96 + 2.86i)19-s + ⋯
L(s)  = 1  + (1.52 − 0.882i)2-s + (−0.816 − 1.41i)3-s + (1.05 − 1.83i)4-s + (−2.49 − 1.43i)6-s + (0.623 + 0.360i)7-s − 1.96i·8-s + (−0.831 + 1.44i)9-s + (0.279 − 0.161i)11-s − 3.44·12-s + (−0.388 + 0.921i)13-s + 1.27·14-s + (−0.677 − 1.17i)16-s + (−0.0772 + 0.133i)17-s + 2.93i·18-s + (1.13 + 0.657i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.804 + 0.593i$
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: \((2,\ 325,\ (\ :1/2),\ -0.804 + 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.747953 - 2.27459i\)
\(L(\frac12)\) \(\approx\) \(0.747953 - 2.27459i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.40 - 3.32i)T \)
good2 \( 1 + (-2.16 + 1.24i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.41 + 2.44i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.64 - 0.952i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.926 + 0.534i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.318 - 0.551i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.96 - 2.86i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.90 + 3.30i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.72 + 8.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (0.655 - 0.378i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.232 - 0.133i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.318 - 0.551i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.44iT - 47T^{2} \)
53 \( 1 - 6.99T + 53T^{2} \)
59 \( 1 + (0.641 + 0.370i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.09 - 3.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.01 + 4.04i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.45 - 4.88i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.71iT - 73T^{2} \)
79 \( 1 + 9.31T + 79T^{2} \)
83 \( 1 - 5.11iT - 83T^{2} \)
89 \( 1 + (10.8 - 6.28i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.65 - 2.11i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.60956957718450023644200388015, −11.10041429560829759631707675707, −9.765259141305849385805007211736, −8.086027934887792280552780815090, −6.93784967045033372616283712288, −6.02303007206520616175365430016, −5.30896041762375586769244589500, −4.12209896379259501707295438139, −2.41214273734519342401260817594, −1.43987374147672883783489464617, 3.28194816282593312327091037268, 4.17598626085751582157722544404, 5.24854098412072861084782126689, 5.42412229717834649910326435764, 6.86123774870959434646419986393, 7.78719034558749697235372500759, 9.281938790362022923947035859116, 10.36907500918585910812144394030, 11.32040997031961078290408665436, 11.92191765621933106523598788727

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