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Properties

Label 2-925-185.84-c1-0-7
Degree $2$
Conductor $925$
Sign $0.824 - 0.565i$
Analytic cond. $7.38616$
Root an. cond. $2.71774$
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  + (−2.29 − 1.32i)2-s + (1.01 − 0.585i)3-s + (2.49 + 4.32i)4-s − 3.09·6-s + (−0.0480 + 0.0277i)7-s − 7.91i·8-s + (−0.815 + 1.41i)9-s − 2.95·11-s + (5.06 + 2.92i)12-s + (−0.898 + 0.518i)13-s + 0.146·14-s + (−5.47 + 9.48i)16-s + (1.08 + 0.625i)17-s + (3.73 − 2.15i)18-s + (1.42 + 2.46i)19-s + ⋯
L(s)  = 1  + (−1.61 − 0.934i)2-s + (0.585 − 0.337i)3-s + (1.24 + 2.16i)4-s − 1.26·6-s + (−0.0181 + 0.0104i)7-s − 2.79i·8-s + (−0.271 + 0.470i)9-s − 0.890·11-s + (1.46 + 0.843i)12-s + (−0.249 + 0.143i)13-s + 0.0391·14-s + (−1.36 + 2.37i)16-s + (0.262 + 0.151i)17-s + (0.880 − 0.508i)18-s + (0.325 + 0.564i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.824 - 0.565i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.824 - 0.565i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.527159 + 0.163264i\)
\(L(\frac12)\) \(\approx\) \(0.527159 + 0.163264i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 + (6.08 + 0.0612i)T \)
good2 \( 1 + (2.29 + 1.32i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.01 + 0.585i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.0480 - 0.0277i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.95T + 11T^{2} \)
13 \( 1 + (0.898 - 0.518i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.08 - 0.625i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.42 - 2.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.00iT - 23T^{2} \)
29 \( 1 + 2.76T + 29T^{2} \)
31 \( 1 - 8.78T + 31T^{2} \)
41 \( 1 + (-3.37 - 5.84i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 11.7iT - 43T^{2} \)
47 \( 1 - 10.8iT - 47T^{2} \)
53 \( 1 + (-3.53 - 2.04i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.27 - 3.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.725 - 1.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.07 - 5.23i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.01 + 3.49i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.65iT - 73T^{2} \)
79 \( 1 + (-8.24 - 14.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.7 + 7.34i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.37 - 7.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.40iT - 97T^{2} \)
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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

−10.13425345554055120792425414433, −9.351452565052393593208747000593, −8.453409986250531664953116162443, −7.967497912904473289856947628643, −7.38649569003892062793201024689, −6.17910160842921913438639268894, −4.58742449511915399053861177088, −3.01777087461869462359468435005, −2.54098569441443354307072406181, −1.31716439442922604474463684380, 0.42620638756345551888948276324, 2.10920154944322297939515733100, 3.37742043606174103706946697444, 5.13269351834289953989531361624, 5.86087936658583533030117979449, 7.01267859073744041312836864561, 7.56650559886813131757019402190, 8.509857610941488404870712936355, 8.937622256895797556378672408604, 9.921176200709821696284982820815

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