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Properties

Label 2-414-207.113-c1-0-13
Degree $2$
Conductor $414$
Sign $0.942 + 0.335i$
Analytic cond. $3.30580$
Root an. cond. $1.81818$
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  + (0.998 + 0.0475i)2-s + (−1.73 + 0.0461i)3-s + (0.995 + 0.0950i)4-s + (−1.81 + 1.73i)5-s + (−1.73 − 0.0363i)6-s + (0.955 − 4.95i)7-s + (0.989 + 0.142i)8-s + (2.99 − 0.159i)9-s + (−1.89 + 1.64i)10-s + (1.94 + 1.00i)11-s + (−1.72 − 0.118i)12-s + (3.99 − 0.769i)13-s + (1.19 − 4.90i)14-s + (3.06 − 3.08i)15-s + (0.981 + 0.189i)16-s + (−0.792 + 1.73i)17-s + ⋯
L(s)  = 1  + (0.706 + 0.0336i)2-s + (−0.999 + 0.0266i)3-s + (0.497 + 0.0475i)4-s + (−0.812 + 0.774i)5-s + (−0.706 − 0.0148i)6-s + (0.361 − 1.87i)7-s + (0.349 + 0.0503i)8-s + (0.998 − 0.0532i)9-s + (−0.600 + 0.519i)10-s + (0.585 + 0.301i)11-s + (−0.498 − 0.0342i)12-s + (1.10 − 0.213i)13-s + (0.318 − 1.31i)14-s + (0.791 − 0.796i)15-s + (0.245 + 0.0473i)16-s + (−0.192 + 0.421i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.942 + 0.335i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.942 + 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49863 - 0.258711i\)
\(L(\frac12)\) \(\approx\) \(1.49863 - 0.258711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.998 - 0.0475i)T \)
3 \( 1 + (1.73 - 0.0461i)T \)
23 \( 1 + (-4.79 + 0.0933i)T \)
good5 \( 1 + (1.81 - 1.73i)T + (0.237 - 4.99i)T^{2} \)
7 \( 1 + (-0.955 + 4.95i)T + (-6.49 - 2.60i)T^{2} \)
11 \( 1 + (-1.94 - 1.00i)T + (6.38 + 8.96i)T^{2} \)
13 \( 1 + (-3.99 + 0.769i)T + (12.0 - 4.83i)T^{2} \)
17 \( 1 + (0.792 - 1.73i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (-4.90 + 2.23i)T + (12.4 - 14.3i)T^{2} \)
29 \( 1 + (0.270 + 2.83i)T + (-28.4 + 5.48i)T^{2} \)
31 \( 1 + (6.52 + 5.12i)T + (7.30 + 30.1i)T^{2} \)
37 \( 1 + (-1.82 - 6.21i)T + (-31.1 + 20.0i)T^{2} \)
41 \( 1 + (5.01 + 5.25i)T + (-1.95 + 40.9i)T^{2} \)
43 \( 1 + (-1.00 - 1.27i)T + (-10.1 + 41.7i)T^{2} \)
47 \( 1 + (-1.68 + 0.974i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.59 + 4.14i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (1.41 + 7.32i)T + (-54.7 + 21.9i)T^{2} \)
61 \( 1 + (2.44 + 6.10i)T + (-44.1 + 42.0i)T^{2} \)
67 \( 1 + (-6.69 - 12.9i)T + (-38.8 + 54.5i)T^{2} \)
71 \( 1 + (5.61 - 8.73i)T + (-29.4 - 64.5i)T^{2} \)
73 \( 1 + (-3.76 - 8.25i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (0.829 + 0.287i)T + (62.0 + 48.8i)T^{2} \)
83 \( 1 + (-4.16 - 3.96i)T + (3.94 + 82.9i)T^{2} \)
89 \( 1 + (-2.26 - 15.7i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (12.6 - 3.07i)T + (86.2 - 44.4i)T^{2} \)
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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

−11.12758904968411963227326091104, −10.82002704696848859256513489473, −9.766888219193749908185095444998, −7.927912766928024819669472176606, −7.06768473071059879938836803931, −6.68400092805079015628909179686, −5.27380920281673928038430396085, −4.08164150305355950024521655032, −3.63166142163954012913144179843, −1.11216694595322941866970708373, 1.46715314146502196261434177762, 3.38647317319099281912882889221, 4.67666887258281827292039673797, 5.44361913453059668468821962473, 6.14720285898638717967227456434, 7.39307244291240127260362537869, 8.642366878259596506311325169113, 9.255240420443612248085818246597, 10.93547395084106354126485549133, 11.51004073787734626980725556051

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