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Properties

Label 2-325-65.64-c1-0-7
Degree $2$
Conductor $325$
Sign $0.971 - 0.235i$
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  − 2.51·2-s + 1.51i·3-s + 4.32·4-s − 3.80i·6-s + 3.32·7-s − 5.83·8-s + 0.707·9-s − 2.83i·11-s + 6.54i·12-s + (0.806 − 3.51i)13-s − 8.34·14-s + 6.02·16-s − 6.64i·17-s − 1.77·18-s + 2.19i·19-s + ⋯
L(s)  = 1  − 1.77·2-s + 0.874i·3-s + 2.16·4-s − 1.55i·6-s + 1.25·7-s − 2.06·8-s + 0.235·9-s − 0.854i·11-s + 1.88i·12-s + (0.223 − 0.974i)13-s − 2.23·14-s + 1.50·16-s − 1.61i·17-s − 0.419·18-s + 0.503i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.971 - 0.235i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.971 - 0.235i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.703229 + 0.0840765i\)
\(L(\frac12)\) \(\approx\) \(0.703229 + 0.0840765i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.806 + 3.51i)T \)
good2 \( 1 + 2.51T + 2T^{2} \)
3 \( 1 - 1.51iT - 3T^{2} \)
7 \( 1 - 3.32T + 7T^{2} \)
11 \( 1 + 2.83iT - 11T^{2} \)
17 \( 1 + 6.64iT - 17T^{2} \)
19 \( 1 - 2.19iT - 19T^{2} \)
23 \( 1 - 0.485iT - 23T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 + 3.80iT - 31T^{2} \)
37 \( 1 - 9.32T + 37T^{2} \)
41 \( 1 - 1.61iT - 41T^{2} \)
43 \( 1 - 0.872iT - 43T^{2} \)
47 \( 1 + 3.32T + 47T^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 - 8.83iT - 59T^{2} \)
61 \( 1 + 3.70T + 61T^{2} \)
67 \( 1 + 4.29T + 67T^{2} \)
71 \( 1 + 2.19iT - 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + 0.585T + 79T^{2} \)
83 \( 1 - 7.70T + 83T^{2} \)
89 \( 1 + 3.41iT - 89T^{2} \)
97 \( 1 + 0.641T + 97T^{2} \)
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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

−11.16464300172856585579635594194, −10.56996117927670162517658963082, −9.731744547455279143915691540078, −8.941401121387006481306277464484, −8.026257149342699668783736278677, −7.41644499722193559129017001499, −5.88802135872215528763261607001, −4.61067323617825648152040258097, −2.86177683701127673778854391554, −1.09568076286510034531210624994, 1.41377088203708015654048802902, 2.06540704347526977714218418144, 4.49515536275461632505894386410, 6.35865018412583693881678197803, 7.09391056602195649614593513187, 7.974106809746168214521881359357, 8.527509234218118528304685506081, 9.624674281925363323213756181454, 10.54412543862200740261642450140, 11.36439379411337636089062570440

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