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Properties

Label 1-13e2-169.90-r0-0-0
Degree $1$
Conductor $169$
Sign $0.858 - 0.513i$
Analytic cond. $0.784832$
Root an. cond. $0.784832$
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  + (−0.568 − 0.822i)2-s + (0.885 + 0.464i)3-s + (−0.354 + 0.935i)4-s + (0.748 − 0.663i)5-s + (−0.120 − 0.992i)6-s + (0.970 − 0.239i)7-s + (0.970 − 0.239i)8-s + (0.568 + 0.822i)9-s + (−0.970 − 0.239i)10-s + (−0.568 + 0.822i)11-s + (−0.748 + 0.663i)12-s + (−0.748 − 0.663i)14-s + (0.970 − 0.239i)15-s + (−0.748 − 0.663i)16-s + (−0.970 + 0.239i)17-s + (0.354 − 0.935i)18-s + ⋯
L(s)  = 1  + (−0.568 − 0.822i)2-s + (0.885 + 0.464i)3-s + (−0.354 + 0.935i)4-s + (0.748 − 0.663i)5-s + (−0.120 − 0.992i)6-s + (0.970 − 0.239i)7-s + (0.970 − 0.239i)8-s + (0.568 + 0.822i)9-s + (−0.970 − 0.239i)10-s + (−0.568 + 0.822i)11-s + (−0.748 + 0.663i)12-s + (−0.748 − 0.663i)14-s + (0.970 − 0.239i)15-s + (−0.748 − 0.663i)16-s + (−0.970 + 0.239i)17-s + (0.354 − 0.935i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.858 - 0.513i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (90, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ 0.858 - 0.513i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.255177444 - 0.3467657054i\)
\(L(\frac12)\) \(\approx\) \(1.255177444 - 0.3467657054i\)
\(L(1)\) \(\approx\) \(1.154533341 - 0.2696248374i\)
\(L(1)\) \(\approx\) \(1.154533341 - 0.2696248374i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (-0.568 - 0.822i)T \)
3 \( 1 + (0.885 + 0.464i)T \)
5 \( 1 + (0.748 - 0.663i)T \)
7 \( 1 + (0.970 - 0.239i)T \)
11 \( 1 + (-0.568 + 0.822i)T \)
17 \( 1 + (-0.970 + 0.239i)T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + (0.568 + 0.822i)T \)
31 \( 1 + (-0.120 - 0.992i)T \)
37 \( 1 + (-0.120 - 0.992i)T \)
41 \( 1 + (-0.885 - 0.464i)T \)
43 \( 1 + (0.120 - 0.992i)T \)
47 \( 1 + (0.354 + 0.935i)T \)
53 \( 1 + (-0.970 + 0.239i)T \)
59 \( 1 + (0.748 - 0.663i)T \)
61 \( 1 + (-0.970 - 0.239i)T \)
67 \( 1 + (0.354 + 0.935i)T \)
71 \( 1 + (-0.885 - 0.464i)T \)
73 \( 1 + (-0.568 + 0.822i)T \)
79 \( 1 + (-0.354 - 0.935i)T \)
83 \( 1 + (-0.885 + 0.464i)T \)
89 \( 1 - T \)
97 \( 1 + (0.748 + 0.663i)T \)
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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

−27.07256367187747384459672077787, −26.713649012631649334472337992432, −25.6288171077712856748603149176, −24.94439135659410258683811321967, −24.17926053022372539702170643338, −23.23817076741201885175264292928, −21.71044140691703274495472054355, −20.86388399542728178806091543626, −19.50608963482539517750876680093, −18.62194091829730180478837105750, −17.994940822167872613450594060693, −17.0965219222462849758859590764, −15.49996177504685781578271582001, −14.817336512859649395876703939407, −13.90315768741310825096785357421, −13.21128940662883087043605211720, −11.21411009360716389685419254749, −10.20413259588489156075055531022, −8.89388385989720344197989510124, −8.27130727893094485095783365221, −7.06567703031052435180803305778, −6.150200422843951057266860201390, −4.76391406118186235993303536244, −2.744308252896362807838390975990, −1.5676497077340881796570534010, 1.67667297455195896457004097083, 2.45651171061152390021822177131, 4.20820889279635749133480214933, 4.94946153185829079483501658218, 7.28598479533165961570297957554, 8.454072301996706023456210959698, 9.07044469133822180771741231017, 10.22245756422325191291351739664, 10.95925821751974648788953742709, 12.61412285706210611662883312284, 13.3212509841023579702853189250, 14.41569080949223042860903090871, 15.646528941501789822048814610199, 17.01888716496604839980785459497, 17.666502308199167988975635299747, 18.818201144410850290357978038320, 20.04223462896301712100016529936, 20.650917940828855584341480384773, 21.25065453744928984246052081779, 22.13184408945897673253897826585, 23.76885738396188422763895466633, 25.0254944184146535054717440772, 25.70203478667833741786404781568, 26.66496326745887066736964086068, 27.56580504278247827226004940241

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