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Properties

Label 1-344-344.109-r0-0-0
Degree $1$
Conductor $344$
Sign $0.783 - 0.621i$
Analytic cond. $1.59752$
Root an. cond. $1.59752$
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  + (0.733 − 0.680i)3-s + (0.988 + 0.149i)5-s + (−0.5 + 0.866i)7-s + (0.0747 − 0.997i)9-s + (0.900 − 0.433i)11-s + (−0.365 − 0.930i)13-s + (0.826 − 0.563i)15-s + (−0.988 + 0.149i)17-s + (−0.0747 − 0.997i)19-s + (0.222 + 0.974i)21-s + (0.826 + 0.563i)23-s + (0.955 + 0.294i)25-s + (−0.623 − 0.781i)27-s + (0.733 + 0.680i)29-s + (0.955 − 0.294i)31-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)3-s + (0.988 + 0.149i)5-s + (−0.5 + 0.866i)7-s + (0.0747 − 0.997i)9-s + (0.900 − 0.433i)11-s + (−0.365 − 0.930i)13-s + (0.826 − 0.563i)15-s + (−0.988 + 0.149i)17-s + (−0.0747 − 0.997i)19-s + (0.222 + 0.974i)21-s + (0.826 + 0.563i)23-s + (0.955 + 0.294i)25-s + (−0.623 − 0.781i)27-s + (0.733 + 0.680i)29-s + (0.955 − 0.294i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $0.783 - 0.621i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ 0.783 - 0.621i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.767163903 - 0.6161442997i\)
\(L(\frac12)\) \(\approx\) \(1.767163903 - 0.6161442997i\)
\(L(1)\) \(\approx\) \(1.463096822 - 0.3050723183i\)
\(L(1)\) \(\approx\) \(1.463096822 - 0.3050723183i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (0.733 - 0.680i)T \)
5 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (-0.365 - 0.930i)T \)
17 \( 1 + (-0.988 + 0.149i)T \)
19 \( 1 + (-0.0747 - 0.997i)T \)
23 \( 1 + (0.826 + 0.563i)T \)
29 \( 1 + (0.733 + 0.680i)T \)
31 \( 1 + (0.955 - 0.294i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (-0.365 + 0.930i)T \)
59 \( 1 + (-0.623 - 0.781i)T \)
61 \( 1 + (-0.955 - 0.294i)T \)
67 \( 1 + (-0.0747 - 0.997i)T \)
71 \( 1 + (0.826 - 0.563i)T \)
73 \( 1 + (0.365 + 0.930i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.733 - 0.680i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.03049617672127509505491639974, −24.4724985338536976112455341252, −22.94317610897954588887615714611, −22.28582926977580346741412116957, −21.29447094710305681063394838781, −20.67190291495413926312009687805, −19.74486760743982834231878610788, −19.06698700776973170036730224810, −17.614968014163721121175378513847, −16.83966760034871600096000141538, −16.17784947086953531539822983494, −14.87797478002556081461313575360, −14.08935212846206726235748715119, −13.531202657939919574136008943765, −12.4119336667791075333014902505, −10.9668946413641965833885272497, −10.00096359782375495394151240932, −9.43999699204007955721927962188, −8.56553378865343373618386491879, −7.11355022394974385333223406112, −6.29448347763637366842757630832, −4.71775247750983392393861550814, −4.05943184782032772643036656808, −2.69877002477465164686826399087, −1.59344200546787691854188183078, 1.26750887918023844709316650611, 2.56098556385814856582998223314, 3.1529006042412108069282351891, 4.961784642257827008504234630554, 6.294851639013336261350580902826, 6.70532068951256135642979400737, 8.22322036233104041985552460945, 9.105780854188249737872509524227, 9.68718431783132704430741458561, 11.11897119924620516402748117101, 12.296876984204431114290788077626, 13.15606638002298058914152718665, 13.7489240675519404221756871280, 14.89475984322475627635679507723, 15.47687532964106369976856707920, 17.03571345630638186142663183547, 17.77662356980193446336610551190, 18.5644435247895382230157085205, 19.5445695642362524129335721773, 20.111923978909277435596093473434, 21.465862539008104106913182081658, 21.9487329648946796352150759389, 22.96780119636809685805704251531, 24.36462687870202441496763551153, 24.853278999620807232524607049123

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