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Properties

Label 2-3675-735.257-c0-0-1
Degree $2$
Conductor $3675$
Sign $0.904 + 0.426i$
Analytic cond. $1.83406$
Root an. cond. $1.35427$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.999 + 0.0373i)3-s + (0.930 + 0.365i)4-s + (−0.757 − 0.652i)7-s + (0.997 + 0.0747i)9-s + (0.916 + 0.399i)12-s + (−0.631 − 1.80i)13-s + (0.733 + 0.680i)16-s + (0.997 − 1.72i)19-s + (−0.733 − 0.680i)21-s + (0.993 + 0.111i)27-s + (−0.467 − 0.884i)28-s + (−1.35 + 0.781i)31-s + (0.900 + 0.433i)36-s + (−0.119 + 0.273i)37-s + (−0.563 − 1.82i)39-s + ⋯
L(s)  = 1  + (0.999 + 0.0373i)3-s + (0.930 + 0.365i)4-s + (−0.757 − 0.652i)7-s + (0.997 + 0.0747i)9-s + (0.916 + 0.399i)12-s + (−0.631 − 1.80i)13-s + (0.733 + 0.680i)16-s + (0.997 − 1.72i)19-s + (−0.733 − 0.680i)21-s + (0.993 + 0.111i)27-s + (−0.467 − 0.884i)28-s + (−1.35 + 0.781i)31-s + (0.900 + 0.433i)36-s + (−0.119 + 0.273i)37-s + (−0.563 − 1.82i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.904 + 0.426i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.904 + 0.426i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.135127355\)
\(L(\frac12)\) \(\approx\) \(2.135127355\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.999 - 0.0373i)T \)
5 \( 1 \)
7 \( 1 + (0.757 + 0.652i)T \)
good2 \( 1 + (-0.930 - 0.365i)T^{2} \)
11 \( 1 + (0.988 - 0.149i)T^{2} \)
13 \( 1 + (0.631 + 1.80i)T + (-0.781 + 0.623i)T^{2} \)
17 \( 1 + (-0.294 + 0.955i)T^{2} \)
19 \( 1 + (-0.997 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.294 + 0.955i)T^{2} \)
29 \( 1 + (-0.222 - 0.974i)T^{2} \)
31 \( 1 + (1.35 - 0.781i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.119 - 0.273i)T + (-0.680 - 0.733i)T^{2} \)
41 \( 1 + (-0.900 + 0.433i)T^{2} \)
43 \( 1 + (-0.461 - 0.734i)T + (-0.433 + 0.900i)T^{2} \)
47 \( 1 + (0.930 + 0.365i)T^{2} \)
53 \( 1 + (0.680 - 0.733i)T^{2} \)
59 \( 1 + (-0.826 + 0.563i)T^{2} \)
61 \( 1 + (1.73 - 0.680i)T + (0.733 - 0.680i)T^{2} \)
67 \( 1 + (0.352 - 1.31i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.367 - 1.94i)T + (-0.930 + 0.365i)T^{2} \)
79 \( 1 + (-0.632 - 0.365i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.781 - 0.623i)T^{2} \)
89 \( 1 + (0.988 + 0.149i)T^{2} \)
97 \( 1 + (-1.16 + 1.16i)T - iT^{2} \)
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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

−8.602628504579991472709325723765, −7.66152780168906824259127635485, −7.35770018547754599828201479130, −6.78808390514836568229282981135, −5.72583699629776310136035332157, −4.79116508844815462711440094663, −3.65214695198033080555031100938, −3.00461698913159166967063552139, −2.59212601596023274608683847953, −1.12838678788571886789051358942, 1.73811145344380419261659083778, 2.14237434839304910629385923246, 3.22526837457485091820470355935, 3.84591898025161409181888780367, 5.02074842341356178871422159815, 6.01339433451564270987238856700, 6.55352036758216604809156622587, 7.44853351492924382127065816023, 7.76091591054081022385756381845, 9.078605725275843075241022527177

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