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Properties

Label 2-414-9.4-c1-0-2
Degree $2$
Conductor $414$
Sign $-0.966 + 0.256i$
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  + (0.5 + 0.866i)2-s + (−0.376 + 1.69i)3-s + (−0.499 + 0.866i)4-s + (−0.990 + 1.71i)5-s + (−1.65 + 0.518i)6-s + (0.245 + 0.425i)7-s − 0.999·8-s + (−2.71 − 1.27i)9-s − 1.98·10-s + (−1.91 − 3.32i)11-s + (−1.27 − 1.17i)12-s + (−2.84 + 4.92i)13-s + (−0.245 + 0.425i)14-s + (−2.52 − 2.32i)15-s + (−0.5 − 0.866i)16-s + 4.35·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.217 + 0.976i)3-s + (−0.249 + 0.433i)4-s + (−0.442 + 0.766i)5-s + (−0.674 + 0.211i)6-s + (0.0927 + 0.160i)7-s − 0.353·8-s + (−0.905 − 0.424i)9-s − 0.626·10-s + (−0.578 − 1.00i)11-s + (−0.368 − 0.338i)12-s + (−0.789 + 1.36i)13-s + (−0.0656 + 0.113i)14-s + (−0.652 − 0.599i)15-s + (−0.125 − 0.216i)16-s + 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.966 + 0.256i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.966 + 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.131240 - 1.00593i\)
\(L(\frac12)\) \(\approx\) \(0.131240 - 1.00593i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.376 - 1.69i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.990 - 1.71i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.245 - 0.425i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.91 + 3.32i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.84 - 4.92i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.35T + 17T^{2} \)
19 \( 1 - 4.40T + 19T^{2} \)
29 \( 1 + (4.47 + 7.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.82 - 8.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.12T + 37T^{2} \)
41 \( 1 + (1.69 - 2.93i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.76 - 8.25i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.30 - 10.9i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.481T + 53T^{2} \)
59 \( 1 + (1.38 - 2.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.41 - 2.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.245 + 0.425i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.93T + 71T^{2} \)
73 \( 1 - 6.59T + 73T^{2} \)
79 \( 1 + (-6.68 - 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.956 + 1.65i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.17T + 89T^{2} \)
97 \( 1 + (-6.47 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.57240106748223519986057770114, −10.94294397044807154137755651651, −9.832337287995189236657940795953, −9.043496839275839064628413072303, −7.899561496250053459257093306991, −7.02793258859503484024269582984, −5.83056200392164069938105134149, −5.06323566838812446962443332390, −3.81810949445497966069421010249, −2.97270502645439724782294087870, 0.59457158024240716549585809484, 2.12409645534624956083425432119, 3.50132512760822309809313677123, 5.14894993715353035669216236986, 5.46335913766887503972619157071, 7.29705006130070171890421150906, 7.70614401550760149201151554371, 8.875536391757370589847011563178, 10.05651200244491322479156117457, 10.82773318615048800016246407192

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